232 UTL_NLA

The UTL_NLA package exposes a subset of the BLAS and LAPACK (Version 3.0) operations on vectors and matrices represented as VARRAYs.

This chapter contains the following topics:

  • Using UTL_NLA

    • Overview

    • Rules and Limits

    • Security Model

  • Subprogram Groups

    • BLAS Level 1 (Vector-Vector Operations) Subprograms

    • BLAS Level 2 (Matrix-Vector Operations) Subprograms

    • BLAS Level 3 (Matrix-Matrix Operations) Subprograms

    • LAPACK Driver Routines (Linear Equations) Subprograms

    • LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

  • Summary of UTL_NLA Subprograms

Using UTL_NLA

This section contains topics which relate to using the UTL_NLA package.

Overview

The UTL_NLA package exposes a subset of the BLAS (Basic Linear Algebra Subprograms) and LAPACK (Linear Algebra PACKage )(Version 3.0) operations on vectors and matrices represented as VARRAYs.

Standards

For more information on the BLAS and LAPACK standards see


http://www.netlib.org/blas/
http://www.netlib.org/lapack/

Required Expertise

Users of this package are expected to have a sound grasp of linear algebra in general and of the BLAS and LAPACK libraries in particular.

Implementation

The mapping between BLAS and LAPACK procedures and their corresponding PL/SQL calls is one-to-one.

  • All BLAS functions have the BLAS_ prefix (for example, the BLAS_ASUM Functions). The subroutines and functions in BLAS are mapped to PL/SQL procedures and functions, respectively.

  • All LAPACK functions have the LAPACK_ prefix (for example, the LAPACK_GBSV Procedures). The subroutines in LAPACK are mapped to PL/SQL procedures. Procedures that perform the same operation but differ only on the datatype of the arguments have the same overloaded names.

The mapping between BLAS and LAPACK procedure parameters and those of their corresponding PL/SQL subprograms is almost one-to-one.

  • Also in the PL/SQL interface for LAPACK, all /work/ arguments have been removed. The UTL_NLA package manages the allocation and de-allocation of all work areas required by the libraries.

  • A new optional parameter, pack, has been added to the end of each LAPACK procedure that specifies if the matrix has been linearized in the row-major or column-major (default) format.

Rules and Limits

Vectors and matrices are stored in VARRAYs with a maximum size of one million entries. Given this restriction, UTL_NLA vectors can be up to one million entries but matrices need to be of size RxC <= 1,000,000.

Security Model

The UTL_NLA package is owned by user SYS and is installed as part of database installation. Execution privilege on the package is granted to public. The routines in the package are run with invokers' rights (run with the privileges of the current user).

Subprogram Groups

BLAS Level 1 (Vector-Vector Operations) Subprograms

Table 232-1 BLAS Level 1 (Vector-Vector Operations) Subprograms

Subprogram Description

BLAS_ASUM Functions

Computes the sum of the absolute values of the vector components

BLAS_AXPY Procedures

Copies alpha*X + Y into vector Y

BLAS_COPY Procedures

Copies the contents of vector X to vector Y

BLAS_DOT Functions

Returns the dot (scalar) product of two vectors X and Y

BLAS_IAMAX Functions

Computes the index of the first element of a vector that has the largest absolute value

BLAS_NRM2 Functions

Computes the vector 2-norm (Euclidean norm)

BLAS_ROT Procedures

Returns the plane rotation of points

BLAS_ROTG Procedures

Returns the Givens rotation of points

BLAS_SCAL Procedures

Scales a vector by a constant

BLAS_SWAP Procedures

Swaps the contents of two vectors each of size n

BLAS Level 2 (Matrix-Vector Operations) Subprograms

Table 232-2 BLAS Level 2 (Matrix-Vector Operations) Subprograms

Subprogram Description

BLAS_GBMV Procedures

Performs the matrix-vector operation y := alpha*A*x + beta*y or y := alpha*A'*x + beta*y where alpha and beta are scalars, x and y are vectors and A is an m by n band matrix, with kl sub-diagonals and ku super-diagonals

BLAS_GEMV Procedures

Performs the matrix-vector operations y := alpha*A*x + beta*y or y := alpha*A'*x + beta*y where alpha and beta are scalars, x and y are vectors and A is an m by n matrix

BLAS_GER Procedures

Performs a rank 1 operation A := alpha*x*y' + A where alpha is a scalar, x is an m element vector, y is an n element vector and A is an m by n matrix

BLAS_SBMV Procedures

Performs a matrix-vector operation y := alpha*A*x + beta*y where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric band matrix, with k super-diagonals

BLAS_SPMV Procedures

Performs a matrix-vector operation y := alpha*A*x + beta*y where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix, supplied in packed form

BLAS_SPR Procedures

Performs a symmetric rank 1 operation A := alpha*x*x' + A where alpha is a real scalar, x is an n element vector, and A is an n by n symmetric matrix, supplied in packed form

BLAS_SPR2 Procedures

Performs a symmetric rank 2 operation A := alpha*x*y' + alpha*y*x' +A where alpha is a scalar, x and y are n element vectors, and A is an n by n symmetric matrix, supplied in packed form

BLAS_SBMV Procedures

Performs a matrix-vector operation y := alpha*A*x + beta*y where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric band matrix, with k super-diagonals

BLAS_SYMV Procedures

Performs a matrix-vector operation y := alpha*A*x + beta*y where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix

BLAS_SYR Procedures

Performs a symmetric rank 1 operation A := alpha*x*x' + A where alpha is a real scalar, x is an n element vector, and A is an n by n symmetric matrix

BLAS_SYR2 Procedures

Performs a symmetric rank 2 operation A := alpha*x*y' + alpha*y*x' + A where alpha is a scalar, x and y are n element vectors, and A is an n by n symmetric matrix

BLAS_TBMV Procedures

Performs a matrix-vector operation x := A*x or A'*x = b where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with (k + 1) diagonals

BLAS_TBSV Procedures

Solves one of the systems of equation A*x = b or A'*x = b where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with (k + 1) diagonals

BLAS_TPMV Procedures

Performs a matrix-vector operation x := A*x or x := A'*x where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular matrix, supplied in packed form

BLAS_TPSV Procedures

Solves one of the systems of equation A*x = b or A'*x = b where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular matrix, supplied in packed form

BLAS_TRMV Procedures

Performs a matrix-vector operation x := A*x or x := A'*x where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular matrix

BLAS_TRSV Procedures

Solves one of the systems of equation A*x = b or A'*x = b where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular matrix

BLAS Level 3 (Matrix-Matrix Operations) Subprograms

Table 232-3 BLAS Level 3 (Matrix-Matrix Operations) Subprograms

Subprogram Description

BLAS_GEMM Procedures

Performs one of the matrix-vector operations C := alpha*op(A)*op(B) + beta*C where op(X) is one of op(X) = X or op(X) = X' where alpha and beta are scalars, and A, B and C are matrices, with op(A) an m by k matrix, op(B) a k by n matrix and C an m by n matrix

BLAS_SYMM Procedures

Performs one of the matrix-vector operations C := alpha*A*B + beta*C or C := alpha*B*A + beta*C where alpha and beta are scalars, A is a symmetric matrix, and B and C are m by n matrices

BLAS_SYR2K Procedures

Performs one of the symmetric rank2 k operations C := alpha*A*B' + alpha*B*A' + beta*C or C := alpha*A'*B + alpha*B'*A + beta*C where alpha and beta are scalars, C is an n by n symmetric matrix and A and B are n by k matrices in the first case and k by n matrices in the second case

BLAS_SYRK Procedures

Performs one of the symmetric rank k operations C := alpha*A*A' + beta*C or C := alpha*A'*A + beta*C where alpha and beta are scalars, C is an n by n symmetric matrix and A is an n by k matrix in the first case and a k by n matrix in the second case

BLAS_TRMM Procedures

Performs one of the matrix-vector operations B := alpha*op(A)*B or B := alpha*B*op(A) where alpha is a scalar, B is an m by n matrix, A is a unit, or non-unit, upper or lower triangular matrix and op(A) is one of two alternatives

BLAS_TRSM Procedures

Performs one of the matrix-vector operations op(A)*X = alpha*B or X*op(A) = alpha*B where alpha is a scalar, X and B are m by n matrices, A is a unit, or non-unit, upper or lower triangular matrix, op(A) is one of two alternatives. The matrix X is overwritten on B

LAPACK Driver Routines (Linear Equations) Subprograms

Table 232-4 LAPACK Driver Routines (Linear Equations) Subprograms

Subprogram Description

LAPACK_GBSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n matrix and x and b are n by nrhs matrices. The LU decomposition with partial pivoting and row interchanges is used to factor A.

LAPACK_GESV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n matrix and x and b are n by nrhs matrices. The LU decomposition with partial pivoting and row interchanges is used to factor A.

LAPACK_GTSV Procedures

This procedure solves the equation a * x = b where a is an n by n tridiagonal matrix, by Gaussian elimination with partial pivoting.

LAPACK_PBSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric positive definite band matrix and x and b are n by nrhs matrices. The Cholesky decomposition is used to factor A.

LAPACK_POSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric positive definite matrix and x and b are n by nrhs matrices. The Cholesky decomposition is used to factor A.

LAPACK_PPSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric positive definite matrix stored in packed format and x and b are n by nrhs matrices. The Cholesky decomposition is used to factor A.

LAPACK_PTSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric positive definite tridiagonal matrix, and x and b are n by nrhs matrices.

LAPACK_SPSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric matrix stored in packed format, and x and b are n by nrhs matrices. The diagonal pivoting method is used to factor A.

LAPACK_SYSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric matrix, and x and b are n by nrhs matrices. The diagonal pivoting method is used to factor A.

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

Table 232-5 LAPACK Driver Routines (LLS and Eigenvalue Problems)

Subprogram Description

LAPACK_GEES Procedures

Computes for an n by n real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).

LAPACK_GEEV Procedures

Computes for an n by n real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

LAPACK_GELS Procedures

Solves overdetermined or underdetermined real linear systems involving an m by n matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank.

LAPACK_GESDD Procedures

Computes the singular value decomposition (SVD) of a real m by n matrix A, optionally computing the left and right singular vectors. If singular vectors are desired, it uses a divide-and-conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK_GESVD Procedures

Computes the singular value decomposition (SVD) of a real m by n matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * transpose(V).

LAPACK_SBEV Procedures

Computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A

LAPACK_SBEVD Procedures

Ccomputes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK_SPEV Procedures

Computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage

LAPACK_SPEVD Procedures

Computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK_STEV Procedures

Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A

LAPACK_STEVD Procedures

Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK_SYEV Procedures

Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A

LAPACK_SYEVD Procedures

Computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

Summary of UTL_NLA Subprograms

Table 232-6 UTL_NLA Package Subprograms

Subprogram Description Group

BLAS_ASUM Functions

Computes the sum of the absolute values of the vector components

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_AXPY Procedures

Copies alpha*X + Y into vector Y

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_COPY Procedures

Copies the contents of vector X to vector Y

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_DOT Functions

Returns the dot (scalar) product of two vectors X and Y

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_GBMV Procedures

Performs the matrix-vector operation y := alpha*A*x + beta*y or y := alpha*A'*x + beta*y where alpha and beta are scalars, x and y are vectors and A is an m by n band matrix, with kl sub-diagonals and ku super-diagonals

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_GEMM Procedures

Performs one of the matrix-vector operations where alpha and beta are scalars, and A, B and C are matrices, with op(A) an m by k matrix, op(B) a k by n matrix and C an m by n matrix

BLAS Level 3 (Matrix-Matrix Operations) Subprograms

BLAS_GEMV Procedures

Performs the matrix-vector operations y := alpha*A*x + beta*y or y := alpha*A'*x + beta*y where alpha and beta are scalars, x and y are vectors and A is an m by n matrix

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_GER Procedures

Performs a rank 1 operation A := alpha*x*y' + A where alpha is a scalar, x is an m element vector, y is an n element vector and A is an m by n matrix

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_IAMAX Functions

Computes the index of the first element of a vector that has the largest absolute value

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_NRM2 Functions

Computes the vector 2-norm (Euclidean norm)

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_ROT Procedures

Returns the plane rotation of points

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_ROTG Procedures

Returns the Givens rotation of points

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_SBMV Procedures

Performs a matrix-vector operation y := alpha*A*x + beta*y where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric band matrix, with k super-diagonals

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_SCAL Procedures

Scales a vector by a constant

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_SPMV Procedures

Performs a matrix-vector operation y := alpha*A*x + beta*y where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix, supplied in packed form

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_SPR Procedures

Performs a symmetric rank 1 operation A := alpha*x*x' + A where alpha is a real scalar, x is an n element vector, and A is an n by n symmetric matrix, supplied in packed form

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_SPR2 Procedures

Performs a symmetric rank 2 operation where alpha is a scalar, x and y are n element vectors, and A is an n by n symmetric matrix, supplied in packed form

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_SWAP Procedures

Swaps the contents of two vectors each of size n

BLAS Level 1 (Vector-Vector Operations) Subprograms

BLAS_SYMM Procedures

Performs one of the matrix-vector operations where alpha and beta are scalars, A is a symmetric matrix, and B and C are m by n matrices

BLAS Level 3 (Matrix-Matrix Operations) Subprograms

BLAS_SYMV Procedures

Performs a matrix-vector operation where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_SYR Procedures

Performs a symmetric rank 1 operation where alpha is a real scalar, x is an n element vector, and A is an n by n symmetric matrix

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_SYR2 Procedures

Performs a symmetric rank 2 operation where alpha is a scalar, x and y are n element vectors, and A is an n by n symmetric matrix

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_SYR2K Procedures

Performs one of the symmetric rank2 k operations where alpha and beta are scalars, C is an n by n symmetric matrix and A and B are n by k matrices in the first case and k by n matrices in the second case

BLAS Level 3 (Matrix-Matrix Operations) Subprograms

BLAS_SYRK Procedures

Performs one of the symmetric rank k operations where alpha and beta are scalars, C is an n by n symmetric matrix and A is an n by k matrix in the first case and a k by n matrix in the second case

BLAS Level 3 (Matrix-Matrix Operations) Subprograms

BLAS_TBMV Procedures

Performs a matrix-vector operation where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with (k + 1) diagonals

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_TBSV Procedures

Solves one of the systems of equation where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with (k + 1) diagonals

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_TPMV Procedures

Performs a matrix-vector operation where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular matrix, supplied in packed form

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_TPSV Procedures

Solves one of the systems of equation where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular matrix, supplied in packed form

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_TRMM Procedures

Performs one of the matrix-vector operations wherealpha is a scalar, B is an m by n matrix, A is a unit, or non-unit, upper or lower triangular matrix and op(A) is one of two alternatives

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_TRMV Procedures

Performs a matrix-vector operation where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular matrix

BLAS Level 2 (Matrix-Vector Operations) Subprograms

BLAS_TRSM Procedures

Performs one of the matrix-vector operations op( A )*X = alpha*B or X*op(A) = alpha*B where alpha is a scalar, X and B are m by n matrices, A is a unit, or non-unit, upper or lower triangular matrix, op(A) is one of two alternatives. The matrix X is overwritten on B

BLAS Level 3 (Matrix-Matrix Operations) Subprograms

BLAS_TRSV Procedures

Solves one of the systems of equation where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular matrix

BLAS Level 2 (Matrix-Vector Operations) Subprograms

LAPACK_GBSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n matrix and x and b are n by nrhs matrices. The LU decomposition with partial pivoting and row interchanges is used to factor A.

LAPACK Driver Routines (Linear Equations) Subprograms

LAPACK_GEES Procedures

Computes for an n by n real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_GEEV Procedures

Computes for an n by n real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_GELS Procedures

Solves overdetermined or underdetermined real linear systems involving an m by n matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank.

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_GESDD Procedures

Computes the singular value decomposition (SVD) of a real m by n matrix A, optionally computing the left and right singular vectors. If singular vectors are desired, it uses a divide-and-conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_GESV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n matrix and x and b are n by nrhs matrices. The LU decomposition with partial pivoting and row interchanges is used to factor A.

LAPACK Driver Routines (Linear Equations) Subprograms

LAPACK_GESVD Procedures

Computes the singular value decomposition (SVD) of a real m by n matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * transpose(V).

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_GTSV Procedures

This procedure solves the equation a * x = b where a is an n by n tridiagonal matrix, by Gaussian elimination with partial pivoting.

LAPACK Driver Routines (Linear Equations) Subprograms

LAPACK_PBSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric positive definite band matrix and x and b are n by nrhs matrices. The Cholesky decomposition is used to factor A.

LAPACK Driver Routines (Linear Equations) Subprograms

LAPACK_POSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric positive definite matrix and x and b are n by nrhs matrices. The Cholesky decomposition is used to factor A.

LAPACK Driver Routines (Linear Equations) Subprograms

LAPACK_PPSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric positive definite matrix stored in packed format and x and b are n by nrhs matrices. The Cholesky decomposition is used to factor A.

LAPACK Driver Routines (Linear Equations) Subprograms

LAPACK_PTSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric positive definite tridiagonal matrix, and x and b are n by nrhs matrices.

LAPACK Driver Routines (Linear Equations) Subprograms

LAPACK_SBEV Procedures

Computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_SBEVD Procedures

Ccomputes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_SPEV Procedures

Computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_SPEVD Procedures

Computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_SPSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric matrix stored in packed format, and x and b are n by nrhs matrices. The diagonal pivoting method is used to factor A.

LAPACK Driver Routines (Linear Equations) Subprograms

LAPACK_STEV Procedures

Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_STEVD Procedures

Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_SYEVD Procedures

Computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms

LAPACK_SYSV Procedures

This procedure computes the solution to a real system of linear equations a * x = b where a is an n by n symmetric matrix, and x and b are n by nrhs matrices. The diagonal pivoting method is used to factor A.

LAPACK Driver Routines (Linear Equations) Subprograms

BLAS_ASUM Functions

This procedure computes the sum of the absolute values of the vector components.

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_ASUM (
   n      IN       POSITIVEN,
   x      IN       UTL_NLA_ARRAY_DBL,
   incx   IN       POSITIVEN)
 RETURN BINARY_DOUBLE;

UTL_NLA.BLAS_ASUM (
   n      IN       POSITIVEN,
   alpha  IN       SCALAR_DOUBLE,
   x      IN       UTL_NLA_ARRAY_FLT)
 RETURN BINARY_FLOAT

Parameters

Table 232-7 BLAS_ASUM Function Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incx ) )

incx

Specifies the increment for the elements of x. incx must not be zero.

BLAS_AXPY Procedures

This procedure copies alpha*X + Y into vector Y.

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_AXPY (
   n      IN       POSITIVEN,
   alpha  IN       SCALAR_DOUBLE,
   x      IN       UTL_NLA_ARRAY_DBL,
   incx   IN       POSITIVEN,
   y      IN OUT   UTL_NLA_ARRAY_DBL,
   incy   IN       POSITIVEN);

UTL_NLA.BLAS_AXPY (
   n      IN       POSITIVEN,
   alpha  IN       SCALAR_DOUBLE,
   x      IN       UTL_NLA_ARRAY_FLT,
   incx   IN       POSITIVEN,
   y      IN OUT   UTL_NLA_ARRAY_FLT,
   incy   IN       POSITIVEN);

Parameters

Table 232-8 BLAS_AXPY Procedure Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.  

alpha

Specifies the scalar alpha.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incx ) )

incx

Specifies the increment for the elements of x. incx must not be zero.

y

UTL_NLA_ARRAY_FLT/DBL of DIMENSION at least

( 1 + ( n - 1 )*abs( incy) )

incy

Specifies the increment for the elements of y. incy must not be zero.

BLAS_COPY Procedures

This procedure copies the contents of vector X to vector Y.

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_COPY (
   n     IN     POSITIVEN,
   x     IN     UTL_NLA_ARRAY_DBL,
   incx  IN     POSITIVEN,
   y     IN OUT UTL_NLA_ARRAY_DBL,
   incy  IN     POSITIVEN);

UTL_NLA.BLAS_COPY (
   n     IN     POSITIVEN,
   x     IN     UTL_NLA_ARRAY_FLT,
   incx  IN     POSITIVEN,
   y     IN OUT UTL_NLA_ARRAY_FLT,
   incy  IN     POSITIVEN);

Parameters

Table 232-9 BLAS_COPY Procedure Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incx ) )

incx

Specifies the increment for the elements of x. incx must not be zero.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incy) )

incy

Specifies the increment for the elements of y. incy must not be zero.

BLAS_DOT Functions

This function returns the dot (scalar) product of two vectors X and Y.

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_DOT (
   n     IN   POSITIVEN,
   x     IN   UTL_NLA_ARRAY_DBL,
   incx  IN   POSITIVEN,
   y     IN   UTL_NLA_ARRAY_DBL,
   incy  IN   POSITIVEN) 
  RETURN BINARY_DOUBLE;

UTL_NLA.BLAS_DOT (
   n     IN   POSITIVEN,
   x     IN   UTL_NLA_ARRAY_FLT,
   incx  IN   POSITIVEN,
   y     IN   UTL_NLA_ARRAY_FLT,
   incy  IN   POSITIVEN) 
  RETURN BINARY_FLOAT;

Parameters

Table 232-10 BLAS_DOT Function Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

 ( 1 + ( n - 1 )*abs( incx ) )

incx

Specifies the increment for the elements of x. incx must not be zero.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incy) )

incy

Specifies the increment for the elements of y. incy must not be zero.

BLAS_GBMV Procedures

This procedure performs one of the matrix-vector operations

y := alpha*A*x + beta*y

or

y := alpha*A'*x + beta*y

where alpha and beta are scalars, x and y are vectors and A is an m by n band matrix, with kl sub-diagonals and ku super-diagonals.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_GEMV (
   trans  IN      flag,
   m      IN      POSITIVEN,   n      IN      POSITIVEN,
   kl     IN      NATURALN,
   ku     IN      NATURALN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   x      IN      UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   y      IN OUT  UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_GEMV (
   trans  IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   kl     IN      NATURALN,
   ku     IN      NATURALN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   x      IN      UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   y      IN OUT  UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-11 BLAS_GBMV Procedure Parameters

Parameter Description

trans

Specifies the operation to be performed:

  • trans = 'N' or 'n'y := alpha*A*x + beta*y

  • trans = 'T' or 't'y := alpha*A'*x + beta*y

  • trans = 'C' or 'c'y := alpha*A'*x + beta*y

m

Specifies the number of rows of the matrix A. m must be at least zero.

n

Specifies the number of columns of the matrix A. n must be at least zero.

kl

Specifies the number of sub-diagonals of the matrix A. kl must satisfy 0. le. kl.

ku

Specifies the number of super-diagonals of the matrix A. ku must satisfy 0 .le. ku.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,n).

Before entry, the leading (kl + ku + 1) by n part of the array A must contain the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row (ku+1) of the array, the first super-diagonal starting at position 2 in row ku, the first sub-diagonal starting at position 1 in row(ku+2), and so on.

Elements in the array A that do not correspond to elements in the band matrix (such as the top left ku by ku triangle) are not referenced.

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least (kl+ku+1).

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incx ) )

when trans = ''N' or 'n' and at least

( 1 + ( m - 1 )*abs( incx ) )

otherwise. Before entry, the incremented array X must contain the vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta. When beta is supplied as zero then y need not be set on input.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( m - 1 )*abs( incy ) )

when trans = 'N' or 'n' and at least

(1+(n-1)*abs(incy))

otherwise. Before entry with beta nonzero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y.

incy

Specifies the increment for the elements of y. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_GEMM Procedures

This procedure performs one of the matrix-matrix operations

C := alpha*op( A )*op( B ) + beta*C

where op(X) is one of

op(X) = X

or

op(X) = X'

where alpha and beta are scalars, and A, B and C are matrices, with op(A) an m by k matrix, op(B) a k by n matrix and C an m by n matrix.

See Also:

BLAS Level 3 (Matrix-Matrix Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_GEMM (
   transa IN      flag,
   transb IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   k      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   b      IN      UTL_NLA_ARRAY_DBL,
   ldb    IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   c      IN OUT  UTL_NLA_ARRAY_DBL,
   ldc    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_GEMM (
   transa IN      flag,
   transb IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   k      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   b      IN      UTL_NLA_ARRAY_FLT,
   ldb    IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   c      IN OUT  UTL_NLA_ARRAY_FLT,
   ldc    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-12 BLAS_GEMM Procedure Parameters

Parameter Description

transa

Specifies the form of op(A) to be used in the matrix multiplication as follows:

  • transa = 'N' or 'n' : op(A) = 'A'

  • transa = 'T' or 't' : op(A) = 'A'

  • transa = 'C' or 'c' : op(A) = 'A'

transb

Specifies the form of op(B) to be used in the matrix multiplication as follows:

  • transb = 'N' or 'n' : op(B) = B

  • transb ='T' or 't' : op(B) = B'

  • transb = 'C' or 'c': op(B) = B'

m

Specifies the number of rows of the matrix op(A) and of the matrix C. m must be at least zero.

n

Specifies the number of columns of the matrix op(B) and of the matrix C. n must be at least zero.

k

Specifies the rows of the matrix op(A) and the number of columns of the matrix op(B). k must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, ka) where ka is k when transa = 'N' or 'n', and is m otherwise. Before entry with transa = 'N' or 'n', the leading m by k part of the array A must contain the matrix A, otherwise the leading k by m part of the array A must contain the matrix A.

lda

Specifies the first dimension of a as declared in the calling (sub) program. When transa = 'N' or 'n', lda must be at least max (1,k).

b

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, kb) where kb is n when transb = ''N' or 'n', and is k otherwise. Before entry with transb = 'N' or 'n', the leading k by n part of the array b must contain the matrix B, otherwise the leading n by k part of the arrayb must contain the matrix B.

ldb

Specifies the first dimension of b as declared in the calling (sub) program. When transb = 'N' or 'n', ldb must be at least max (1, n).

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta. When beta is supplied as zero then c need not be set on input.

c

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (ldc, n). Before entry, the leading m by n part of the array C must contain the matrix C, except when beta is zero, in which case C need not be set on entry. On exit, the arrayC is overwritten by the m by n matrix (alpha*op(A)*op(B) + beta*C).

ldc

Specifies the first dimension of C as declared in the calling (sub) program. ldc must be at least max(1, m).

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_GEMV Procedures

This procedure performs one of the matrix-vector operations

y := alpha*A*x + beta*y

or

y := alpha*A'*x + beta*y

where alpha and beta are scalars, x and y are vectors and A is an m by n matrix.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_GEMV (
   trans  IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   x      IN      UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   y      IN OUT  UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_GEMV (
   trans  IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   x      IN      UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   y      IN OUT  UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-13 BLAS_GEMV Procedure Parameters

Parameter Description

trans

Specifies the operation to be performed:

  • trans = 'N' or 'n',y := alpha*A*x + beta*y

  • trans = 'T' or 't'y := alpha*A'*x + beta*y

  • trans = 'C' or 'c'y := alpha*A'*x + beta*y

m

Specifies the number of rows of the matrix A. m must be at least zero.

n

Specifies the number of columns of the matrix A. n must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, n). Before entry, the leading m by n part of the array a must contain the matrix of coefficients.

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least max(1, m).

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incx ) )

when trans = ''N' or 'n' and at least

(1+(m-1)*abs(incx))

otherwise. Before entry, the incremented array X must contain the vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta. When beta is supplied as zero then y need not be set on input.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( m - 1 )*abs( incy ) )

when trans = 'N' or 'n' and at least

( 1 + ( n - 1 )*abs( incy ) )

otherwise. Before entry with beta nonzero, the incremented array Y must contain the vector y. On exit, Y is overwritten by the updated vector y.

incy

Specifies the increment for the elements of y. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_GER Procedures

This procedure performs the rank 1 operation

A := alpha*x*y' + A

where alpha is a scalar, x is an m element vector, y is an n element vector and A is an m by n matrix.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_GER (
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DBL,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   y      IN      UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_GER (
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLT,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   y      IN      UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-14 BLAS_GER Procedure Parameters

Parameter Description

m

Specifies the number of rows of the matrix A. m must be at least zero.

n

Specifies the number of columns of the matrix A. n must be at least zero.

alpha

Specifies the scalar alpha.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( m - 1 )*abs( incx ) )

Before entry, the incremented array X must contain the m element vector x.

incx

Specifies the increment for the elements of x. incx must not be zero.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incy ) )

Before entry, the incremented array Y must contain the m element vector y.

incy

Specifies the increment for the elements of y. incx must not be zero.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, n).

Before entry, the leading m by n part of the array a must contain the matrix of coefficients. On exit, a is overwritten by the updated matrix.

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least

max( 1, m )

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_IAMAX Functions

This function computes the index of first element of a vector that has the largest absolute value.

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_IAMAX (
   n     IN   POSITIVEN,
   x     IN   UTL_NLA_ARRAY_DBL,
   incx  IN   POSITIVEN,
  RETURN POSITIVEN;

UTL_NLA.BLAS_IAMAX (
   n     IN   POSITIVEN,
   x     IN   UTL_NLA_ARRAY_FLT,
   incx  IN   POSITIVEN,
  RETURN POSITIVEN;

Parameters

Table 232-15 BLAS_IAMAX Function Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.

x

UTL_NLA_ARRAY_FLT/DBL of DIMENSION at least

( 1 + ( n - 1 )*abs( incx ) )

incx

Specifies the increment for the elements of x. incx must not be zero.

BLAS_NRM2 Functions

This function computes the vector 2-norm (Euclidean norm).

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_NRM2 (
   n     IN   POSITIVEN,
   x     IN   UTL_NLA_ARRAY_DBL,
   incx  IN   POSITIVEN)
  RETURN BINARY_DOUBLE;

UTL_NLA.BLAS_NRM2 (
   n     IN   POSITIVEN,
   x     IN   UTL_NLA_ARRAY_FLT,
   incx  IN   POSITIVEN)
  RETURN BINARY_FLOAT;

Parameters

Table 232-16 BLAS_NRM2 Function Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incx ) )

incx

Specifies the increment for the elements of x. incx must not be zero.

BLAS_ROT Procedures

This procedure returns the plane rotation of points.

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_ROT (
   n      IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   y      IN OUT  UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   c      IN      SCALAR_DOUBLE,
   s      IN      SCALAR_DOUBLE);

UTL_NLA.BLAS_ROT (
   n      IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   y      IN OUT  UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   c      IN      SCALAR_DOUBLE,
   s      IN      SCALAR_DOUBLE);

Parameters

Table 232-17 BLAS_ROT Procedure Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)* abs(incx))

incx

Specifies the increment for the elements of x. incx must not be zero.

y

UTL_NLA_ARRAY_FLT/DBL of DIMENSION at least

(1+(n-1)*abs(incy))

incy

Specifies the increment for the elements of y. incy must not be zero.

c

SCALAR_FLOAT/DOUBLE.Specifies the scalar C.

s

SCALAR_FLOAT/DOUBLE.Specifies the scalar S.

BLAS_ROTG Procedures

This procedure returns the Givens rotation of points.

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_ROTG (
   a   IN OUT   SCALAR_DOUBLE,
   b   IN OUT   SCALAR_DOUBLE,
   c   IN OUT   SCALAR_DOUBLE,
   s   IN OUT   SCALAR_DOUBLE);

UTL_NLA.BLAS_ROTG (
   a   IN OUT   SCALAR_FLOAT,
   b   IN OUT   SCALAR_FLOAT,
   c   IN OUT   SCALAR_FLOAT,
   s   IN OUT   SCALAR_FLOAT);

Parameters

Table 232-18 BLAS_ROTG Procedure Parameters

Parameter Description

a

SCALAR_FLOAT/DOUBLE. Specifies the scalar A.

b

SCALAR_FLOAT/DOUBLE. Specifies the scalar B. 

c

SCALAR_FLOAT/DOUBLE. Specifies the scalar C.

s

SCALAR_FLOAT/DOUBLE. Specifies the scalar S. 

BLAS_SCAL Procedures

This procedure scales a vector by a constant.

See Also:

BLAS Level 1 (Vector-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SCAL (
   n      IN  POSITIVEN,
   alpha  IN  SCALAR_DOUBLE,
   x      IN  OUT UTL_NLA_ARRAY_DBL,
   incx   IN  POSITIVEN);

UTL_NLA.BLAS_SCAL (
   n      IN  POSITIVEN,
   alpha  IN  SCALAR_FLOAT,
   x      IN  OUT UTL_NLA_ARRAY_FLT,
   incx   IN  POSITIVEN);

Parameters

Table 232-19 BLAS_SCAL Procedure Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.

alpha

Specifies the scalar alpha.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incx))

incx

Specifies the increment for the elements of x. incx must not be zero.

BLAS_SPMV Procedures

This procedure performs the matrix-vector operation

y := alpha*A*x + beta*y

where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix, supplied in packed form.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SPMV (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   ap     IN      UTL_NLA_ARRAY_DBL,
   x      IN      UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   y      IN OUT  UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SPMV (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   ap      IN     UTL_NLA_ARRAY_FLT,
   x      IN      UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   y      IN OUT  UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-20 BLAS_SPMV Procedure Parameters

Parameter Description

uplo

Specifies the upper or lower triangular part of the matrix A is supplied in the packed array AP:

  • uplo = 'U' or 'u'. The upper triangular part of A is supplied in AP.

  • uplo = 'L' or 'l'. The lower triangular part of A is supplied in AP.

n

Specifies the order of the matrix A. n must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

ap

UTL_NLA_ARRAY_FLT/DBL of dimension at least

((n*(n+1))/2)

Before entry with uplo = 'U' or 'u', the array ap must contain the upper triangular part of the symmetric matrix packed sequentially, column by column, so that ap(1) contains a(1,1), ap(2) and ap(3) contain a(1,2) and a(2,2) respectively, and so on.

Before entry with uplo = 'L' or 'l', the array ap must contain the lower triangular part of the symmetric matrix packed sequentially, column by column, so that ap(1) contains, ap(2) and ap(3) contain a(2,1) and a(3,1) respectively, and so on.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incx))

Before entry, the incremented array X must contain the n element vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta. When beta is supplied as zero then Y need not be set on input.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at leasT

(1+(n-1)*abs(incy))

Before entry, the incremented array Y must contain the n element vector y. On exit, Y is overwritten by the updated vector y.

incy

Specifies the increment for the elements of y. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SPR Procedures

This procedure performs the rank 1 operation

A := alpha*x*x' + A

where alpha is a real scalar, x is an n element vector, and A is an n by n symmetric matrix, supplied in packed form.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SPR (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DBL,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   ap     IN OUT  UTL_NLA_ARRAY_DBL,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SPR (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLT,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   ap     IN OUT  UTL_NLA_ARRAY_FLT,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-21 BLAS_SPR Procedure Parameters

Parameter Description

uplo

Specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array ap:

  • uplo = 'U' or 'u': The upper triangular part of A is supplied in ap.

  • uplo = 'L' or 'l' : The lower triangular part of A is supplied in ap.

n

Specifies the order of the matrix A. n must be at least zero.

alpha

Specifies the scalar alpha.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incx))

Before entry, the incremented array X must contain the m element vector x.

incx

Specifies the increment for the elements of x. incx must not be zero.

ap

UTL_NLA_ARRAY_FLT/DBL of dimension at least

((n*(n +1))/2)

Before entry with uplo = 'U' or 'u', the array ap must contain the upper triangular part of the symmetric matrix packed sequentially, column by column, so that ap(1) contains a(1,1), ap(2) and ap(3) contain a(1,2) and a(2,2) respectively, and so on. On exit, the array ap is overwritten by the upper triangular part of the updated matrix.

Before entry with uplo = 'L' or 'l', the array ap must contain the lower triangular part of the symmetric matrix packed sequentially, column by column, so that ap(1) contains a(1,1), ap(2) and ap(3) contain a(2,1) and a(3,1) respectively, and so on. On exit, the array ap is overwritten by the lower triangular part of the updated matrix

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SPR2 Procedures

This procedure performs the rank 2 operation

A := alpha*x*y' + alpha*y*x' +A

where alpha is a scalar, x and y are n element vectors, and A is an n by n symmetric matrix, supplied in packed form.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SPR2 (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DBL,
   x      IN      UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   y      IN      UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SPR2 (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLT,
   x      IN      UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   y      IN      UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-22 BLAS_SPR2 Procedure Parameters

Parameter Description

uplo

Specifies whether the upper or lower triangular part of the matrix A is supplied in the packed array ap :

  • uplo = 'U' or 'u' : The upper triangular part of A is supplied in ap.

  • uplo = 'L' or 'l' : The lower triangular part of A is supplied in ap.

n

Specifies the order of the matrix A. n must be at least zero.

alpha

Specifies the scalar alpha.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incx))

Before entry, the incremented array X must contain the m element vector x.

incx

Specifies the increment for the elements of x. incx must not be zero.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incy))

Before entry, the incremented array X must contain the m element vector y.

incy

Specifies the increment for the elements of y. incy must not be zero.

ap

UTL_NLA_ARRAY_FLT/DBL of dimension at least

((n*(n+1))/2)

Before entry with uplo = 'U' or 'u', the array ap must contain the upper triangular part of the symmetric matrix packed sequentially, column by column, so that ap(1) contains ap(1) contains a(1,1), ap(2) and ap(3) contain a(1,2) and a(2,2) respectively, and so on. On exit, the array ap is overwritten by the upper triangular part of the updated matrix.

Before entry with uplo = 'L' or 'l', the array ap must contain the lower triangular part of the symmetric matrix packed sequentially, column by column, so that ap(1) contains a(1,1), ap(2) and ap(3) contain a(2,1) and a(3,1) respectively, and so on. On exit, the array ap is overwritten by the lower triangular part of the updated matrix

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least (k + 1).

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SBMV Procedures

This procedure performs the matrix-vector operation

y := alpha*A*x + beta*y

where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric band matrix, with k super-diagonals.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SBMV (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   k      IN      NATURALN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   x      IN      UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   y      IN OUT  UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SBMV (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   k      IN      NATURALN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   x      IN      UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   y      IN OUT  UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-23 BLAS_SBMV Procedure Parameters

Parameter Description

uplo

Specifies whether the upper or lower triangular part of the band matrix A is being supplied:

  • uplo = 'U' or 'u'. The upper triangular part of A is supplied.

  • uplo = 'L' or 'l'. The lower triangular part of A is supplied.

n

Specifies the order of the matrix A. n must be at least zero.

k

Specifies the number of super-diagonals of the matrix A. k must satisfy 0 .le. k.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,n).

Before entry with uplo = 'U' or 'u', the leading (k+1) by n part of the array A must contain the upper triangular band part of the symmetric matrix, supplied column by column, with the leading diagonal of the matrix in row (k+1) of the array, the first super-diagonal starting at position 2 in rowk, and so on. The top left k by k triangle of the array A is not referenced.

Before entry with uplo = 'L' or 'l', the leading (k+1) by n part of the array A must contain the lower triangular band part of the symmetric matrix, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced.

Unchanged on exit

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least (k + 1).

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incx))

Before entry, the incremented array X must contain the n element vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incy))

Before entry, the incremented array Y must contain the n element vector y. On exit, Y is overwritten by the updated vector y.

incy

Specifies the increment for the elements of y. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SWAP Procedures

This procedure swaps the contents of two vectors each of size n.

Syntax

UTL_NLA.BLAS_SWAP (
   n     IN     POSITIVEN,
   x     IN OUT UTL_NLA_ARRAY_DBL,
   incx  IN     POSITIVEN,
   y     IN OUT UTL_NLA_ARRAY_DBL,
   incy  IN     POSITIVEN);

UTL_NLA.BLAS_SWAP (
   n     IN     POSITIVEN,
   x     IN OUT UTL_NLA_ARRAY_FLT,
   incx  IN     POSITIVEN,
   y     IN OUT UTL_NLA_ARRAY_FLT,
   incy  IN     POSITIVEN);

Parameters

Table 232-24 BLAS_SWAP Procedure Parameters

Parameter Description

n

Specifies the number of elements of the vectors x and y. n must be at least zero.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incx))

incx

Specifies the increment for the elements of x. incx must not be zero.

y

UTL_NLA_ARRAY_FLT/DBL of DIMENSION at least

(1+(n-1)*abs(incy))

incy

Specifies the increment for the elements of y. incy must not be zero.

BLAS_SYMM Procedures

This procedure performs one of the matrix-matrix operations

C := alpha*A*B + beta*C

or

C := alpha*B*A + beta*C

where alpha and beta are scalars, A is a symmetric matrix, and B and C are m by n matrices.

See Also:

BLAS Level 3 (Matrix-Matrix Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SYMM (
   side   IN      flag,
   uplo   IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   b      IN      UTL_NLA_ARRAY_DBL,
   ldb    IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   c      IN OUT  UTL_NLA_ARRAY_DBL,
   ldc    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SYMM (
   side   IN      flag,
   uplo   IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   b      IN      UTL_NLA_ARRAY_FLT,
   ldb    IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   c      IN OUT  UTL_NLA_ARRAY_FLT,
   ldc    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-25 BLAS_SYMM Procedure Parameters

Parameter Description

side

Specifies whether the symmetric matrix A appears on the left or right in the operation:

  • side = 'L' or 'l' : C := alpha*A*B + beta*C

  • side = 'R' or 'r' : C := alpha*B*A + beta*C

uplo

Specifies whether the upper or lower triangular part of the array A is to be referenced:

  • uplo = 'U' or 'u' : Only the upper triangular part of the symmetric matrix is to be referenced.

  • uplo = 'L' or 'l' : Only the lower triangular part of the symmetric matrix is to be referenced.

m

Specifies the number of rows of the matrix C. m must be at least zero.

n

Specifies the number of columns of the matrix C. n must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,ka) where ka is m when side = 'L' or 'l', and is n otherwise.

Before entry with side = 'L' or 'l', the leading m by m part of the array A must contain the symmetric matrix, such that when uplo = 'U' or 'u', the leading m by m upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced, and when uplo = 'L' or 'l', the leading m by m lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced.

Before entry with side = 'R' or 'r', the n by n part of the array A must contain the symmetric matrix, such that when uplo = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced, and when uplo = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced.

lda

Specifies the first dimension of a as declared in the calling (sub) program. When side = 'L' or 'l', lda must be at least max(1,m), otherwise lda must be at least max(1,n).

b

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (ldb,n).

Before entry, the leading m by n part of the array B must contain the matrix B.

ldb

Specifies the first dimension of b as declared in the calling (sub) program. ldb must be at least max(1,m).

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta. When beta is supplied as zero then c need not be set on input.

c

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (ldc,n). Before entry, the leading m by n part of the array C must contain the matrix C, except when beta is zero, in which case C need not be set on entry. On exit, the array C is overwritten by the m by n updated matrix.

ldc

Specifies the first dimension of C as declared in the calling (sub) program. ldc must be at least max (1,m).

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SYMV Procedures

This procedure performs the matrix-vector operation

y := alpha*A*x + beta*y

where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SYMV (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   x      IN      UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   y      IN OUT  UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SYMV (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   x      IN      UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   y      IN OUT  UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-26 BLAS_SYMV Procedure Parameters

Parameter Description

uplo

Specifies whether the upper or lower triangular part of the array A is to be referenced:

  • uplo = 'U'or 'u'. Only the upper triangular part ofA is to be referenced.

  • uplo = 'L' or 'l'. Only the lower triangular part of A is to be referenced.

n

Specifies the order of the matrix A. n must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,n). Before entry with uplo = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced.

Before entry with uplo = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced.

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least max(1,n).

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incx))

Before entry, the incremented array X must contain the n element vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta. When beta is supplied as zero then y need not be set on input.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incy))

Before entry, the incremented array Y must contain the n element vector y. On exit, Y is overwritten by the updated vector y.

incy

Specifies the increment for the elements of y. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SYR Procedures

This procedure performs the rank 1 operation

A := alpha*x*x' + A

where alpha is a real scalar, x is an n element vector, and A is an n by n symmetric matrix.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SYR (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DBL,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SYR (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLT,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-27 BLAS_SYR Procedure Parameters

Parameter Description

uplo

Specifies whether the upper or lower triangular part of the array A is to be referenced:

  • uplo = 'U'or 'u' : Only the upper triangular part of A is to be referenced.

  • uplo = 'L'or 'l' : Only the lower triangular part of A is to be referenced.

n

Specifies the order of the matrix A. n must be at least zero.

alpha

Specifies the scalar alpha.  

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1+(n-1)*abs(incx))

Before entry, the incremented array X must contain the m element vector x.

incx

Specifies the increment for the elements of x. incx must not be zero.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, n)

Before entry with uplo = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced. On exit, the upper triangular part of the array A is overwritten by the upper triangular part of the updated matrix.

Before entry with uplo = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced. On exit, the lower triangular part of the array A is overwritten by the lower triangular part of the updated matrix.

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least

max( 1, n )

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SYR2 Procedures

This procedure performs the rank 2 operation

A := alpha*x*y' + alpha*y*x' + A

where alpha is a scalar, x and y are n element vectors, and A is an n by n symmetric matrix.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SYR2 (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DBL,
   x      IN      UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   y      IN      UTL_NLA_ARRAY_DBL,
   incy   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SYR2 (
   uplo   IN      flag,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLT,
   x      IN      UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   y      IN      UTL_NLA_ARRAY_FLT,
   incy   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-28 BLAS_SYR2 Procedure Parameters

Parameter Description

uplo

Specifies whether the upper or lower triangular part of the array A is to be referenced:

  • uplo = 'U' or 'u' : Only the upper triangular part of A is to be referenced.

  • uplo = 'L' or 'l' : Only the lower triangular part of A is to be referenced.

n

Specifies the order of the matrix A. n must be at least zero.  

alpha

Specifies the scalar alpha.  

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incx ) )

Before entry, the incremented array X must contain the m element vector x.

incx

Specifies the increment for the elements of x. incx must not be zero.

y

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 )*abs( incy ) )

Before entry, the incremented array Y must contain the m element vector y.

incy

Specifies the increment for the elements of y. incy must not be zero.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, n)

With uplo = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of A is not referenced. On exit, the upper triangular part of the array A is overwritten by the upper triangular part of the updated matrix.

With uplo = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of A is not referenced. On exit, the lower triangular part of the array A is overwritten by the lower triangular part of the updated matrix.

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least

max( 1, n )

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SYR2K Procedures

This procedure performs one of the symmetric rank2 k operations

C := alpha*A*B' + alpha*B*A' + beta*C

or

C := alpha*A'*B + alpha*B'*A + beta*C

where alpha and beta are scalars, C is an n by n symmetric matrix and A and B are n by k matrices in the first case and k by n matrices in the second case.

See Also:

BLAS Level 3 (Matrix-Matrix Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SYR2K (
   uplo   IN      flag,
   trans  IN      flag,
   n      IN      POSITIVEN,
   k      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   b      IN      UTL_NLA_ARRAY_DBL,
   ldb    IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   c      IN OUT  UTL_NLA_ARRAY_DBL,
   ldc    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SYR2K (
   uplo   IN      flag,
   trans  IN      flag,
   n      IN      POSITIVEN,
   k      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   b      IN OUT  UTL_NLA_ARRAY_FLT,
   ldb    IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   c      IN OUT  UTL_NLA_ARRAY_FLT,
   ldc    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-29 BLAS_SYR2K Procedure Parameters

Parameter Description

uplo

Specifies whether the upper or lower triangular part of the array C is to be referenced:

  • uplo = 'U' or 'u' : Only the upper triangular part of C is to be referenced.

  • uplo = 'L' or 'l' : Only the lower triangular part of C is to be referenced.

trans

Specifies the operations to be performed:

  • trans = 'N' or 'n'C : C := alpha*A*B' + alpha*B*A' + beta*C

  • trans = 'T' or 't'C : C := alpha*A'*B + alpha*B'*A + beta*C

  • trans = 'C' or 'c'C : C := alpha*A'*B + alpha*B'*A + beta*C

n

Specifies the order of matrix C. n must be at least zero.

k

On entry with trans = 'N' or 'n', k specifies the number of columns of the matrices A and B. On entry with trans = 'T' or 't' or trans = 'C' or 'c', k specifies the number of rows of the matrices A and B. k must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,ka) where kb is k when trans = 'N' or 'n', and is n otherwise.

Before entry with trans = 'N' or 'n', the leading n byk part of the array A must contain the matrix A, otherwise the leading k by n part of the array A must contain the matrix A.

lda

Specifies the first dimension of a as declared in the calling (sub) program. When trans = 'N' or 'n', lda must be at leastmax(1,n), otherwise lda must be at least max(1,k).

b

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,kb) where kb is k when trans = 'N' or 'n', and is n otherwise.

Before entry with trans = 'N' or 'n', the leading n byk part of the array B must contain the matrix B, otherwise the leading k by n part of the array B must contain the matrix B.

ldb

Specifies the first dimension of b as declared in the calling (sub) program. When trans = 'N' or 'n', ldb must be at least max (1,n), otherwise ldb must be at least max (1,k).

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta.

c

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (ldc,n).

Before entry with uplo = 'U' or 'u', the leading n by n upper triangular part of the array C must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of C is not referenced. On exit, the upper triangular part of the array C is overwritten by the upper triangular part of the updated matrix.

Before entry with uplo = 'L' or 'l', the leading n by n lower triangular part of the array C must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of C is not referenced. On exit, the lower triangular part of the array C is overwritten by the lower triangular part of the updated matrix.

ldc

Specifies the first dimension of C as declared in the calling (sub) program. ldc must be at least max(1,n).

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_SYRK Procedures

This procedure performs one of the symmetric rank k operations

C := alpha*A*A' + beta*C

or

C := alpha*A'*A + beta*C

where alpha and beta are scalars, C is an n by n symmetric matrix andA is an n by k matrix in the first case and a k byn matrix in the second case.

See Also:

BLAS Level 3 (Matrix-Matrix Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_SYRK (
   uplo   IN      flag,
   trans  IN      flag,
   n      IN      POSITIVEN,
   k      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   beta   IN      SCALAR_DOUBLE,
   c      IN OUT  UTL_NLA_ARRAY_DBL,
   ldc    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_SYRK (
   uplo   IN      flag,
   trans  IN      flag,
   n      IN      POSITIVEN,
   k      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   beta   IN      SCALAR_FLOAT,
   c      IN OUT  UTL_NLA_ARRAY_DBL,
   ldc    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-30 BLAS_SYRK Procedure Parameters

Parameter Description

uplo

Specifies whether the upper or lower triangular part of the array C is to be referenced:

  • uplo = 'U' or 'u' : Only the upper triangular part of C is to be referenced.

  • uplo = 'L' or 'l' : Only the lower triangular part of C is to be referenced.

trans

Specifies the operations to be performed:

  • trans = 'N' or 'n' : C := alpha*A*A' + beta*C

  • trans = 'T' or 't' : C := alpha*A'*A + beta*C

  • trans ='C' or 'c' : C := alpha*A'*A + beta*C

n

Specifies the order of matrix C. n must be at least zero.

k

On entry with trans = 'N' or 'n', k specifies the number of columns of the matrix A. On entry with trans = 'T' or 't' or trans = 'C' or 'c', k specifies the number of rows of the matrix A. k must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,ka) where ka is k when trans = 'N' or 'n', and is n otherwise.

Before entry with trans = 'N' or 'n', the leading n byk part of the array A must contain the matrix A, otherwise the leading k by n part of the array A must contain the matrix A.

lda

Specifies the first dimension of a as declared in the calling (sub) program. When trans = 'N' or 'n', lda must be at least max(1,n), otherwiselda must be at least max(1,k).

beta

SCALAR_FLOAT/DOUBLE. Specifies the scalar beta.

c

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (ldc,n).

Before entry with uplo = 'U' or 'u', the leading n by n upper triangular part of the array C must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of C is not referenced. On exit, the upper triangular part of the array C is overwritten by the upper triangular part of the updated matrix.

Before entry with uplo = 'L' or 'l', the leading n by n lower triangular part of the array C must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of C is not referenced. On exit, the lower triangular part of the array C is overwritten by the lower triangular part of the updated matrix.

ldc

Specifies the first dimension of C as declared in the calling (sub) program. ldc must be at least max(1,n).

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_TBMV Procedures

This procedure performs the matrix-vector operations

x := A*x

or

x := A'*x

where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with (k+1) diagonals.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_TBMV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   k      IN      NATURALN,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_TBMV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   k      IN      NATURALN,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-31 BLAS_TBMV Procedure Parameters

Parameter Description

uplo

Specifies whether the matrix is an upper or lower triangular matrix:

  • uplo = 'U' or 'u'. A is an upper triangular matrix.

  • uplo = 'L' or 'l'. A is a lower triangular matrix.

trans

Specifies the operation to be performed:

  • trans = 'N' or 'n'x := A*x

  • trans = 'T' or 't'x := A'*x

  • trans = 'C' or 'c'x := A'*x

diag

Specifies whether or not A is unit triangular:

  • diag = 'U' or 'u'. A is assumed to be unit triangular.

  • diag = 'N' or 'n'. A is not assumed to be unit triangular.

n

Specifies the order of the matrix A. n must be at least zero.  

k

Specifies whether or not A is unit triangular:

  • with uplo = 'U' or 'u', K specifies the number of super-diagonals of the matrix A.

  • with uplo = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A.

K must satisfy 0 .le. k.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION ( lda, n ).

Before entry with uplo = 'U' or 'u', the leading (k+1) by n part of the array A must contain the upper triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row(k+1) of the array, the first super-diagonal starting at position 2 in row k, and so on. The top left k by k triangle of the array A is not referenced.

Before entry with uplo = 'L' or 'l', the leading (k+1) by n part of the array A must contain the lower triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced.

Note that when diag = ''U' or 'u', the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity.

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least (k+1).

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least (1+(n-1)*abs(incx)). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the transformed vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_TBSV Procedures

This procedure solves one of the systems of equations

A*x = b

or

A'*x = b

where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with (k+1) diagonals.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_TBSV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   k      IN      NATURALN,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_STBSV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   k      IN      NATURALN,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-32 BLAS_TBSV Procedure Parameters

Parameter Description

uplo

Specifies whether the matrix is an upper or lower triangular matrix:

  • uplo = ''U' or 'u'. A is an upper triangular matrix.

  • uplo = 'L' or 'l'. A is a lower triangular matrix.

trans

Specifies the equations to be solved:

  • trans = 'N' or 'n : 'A*x = b

  • trans = 'T' or 't': A'*x = b

  • trans = 'C' or 'c': A'*x = b

diag

Specifies whether or not A is unit triangular:

  • diag = 'U' or 'u' : A is assumed to be unit triangular.

  • diag = 'N' or 'n' : A is not assumed to be unit triangular.

n

Specifies the order of the matrix A. n must be at least zero.

k

Specifies whether or not A is unit triangular:

  • with uplo = 'U' or 'u', K specifies the number of super-diagonals of the matrix A.

  • with uplo = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A.

K must satisfy 0 .le. k.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,n).

Before entry with uplo = 'U' or 'u', the leading (k+1) by n part of the array A must contain the upper triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row (k+1) of the array, the first super-diagonal starting at position 2 in row k, and so on. The top left k by k triangle of the array A is not referenced.

Before entry with uplo = 'L' or 'l', the leading (k+1) by n part of the array A must contain the lower triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced.

Note that when diag = 'U' or 'u', the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity.

lda

On entry, lda specifies the first dimension of A as declared in the calling (sub) program. lda must be at least (k+1).

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 ) *abs( incx ) )

Before entry, the incremented array X must contain the n element right-hand side vector b.

On exit, X is overwritten with the solution vector x.

incx

Specifies the increment for the elements of x. incx must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

Usage Notes

No test for singularity or near-singularity is included in this routine. Such tests must be performed before calling this routine.

BLAS_TPMV Procedures

This procedure performs the matrix-vector operations

x := A*x

or

x := A'*x

where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular matrix, supplied in packed form.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_TPMV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   ap     IN      UTL_NLA_ARRAY_DBL,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_TBMV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   ap     IN      UTL_NLA_ARRAY_FLT,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-33 BLAS_TPMV Procedure Parameters

Parameter Description

uplo

Specifies whether the matrix is an upper or lower triangular matrix:

  • uplo = 'U' or 'u'. A is an upper triangular matrix.

  • uplo = 'L' or 'l'. A is a lower triangular matrix.

trans

Specifies the operation to e performed:

  • trans = 'N' or 'n'x := A*x

  • trans = 'T' or 't'x := A'*x

  • trans = 'C' or 'c'x := A'*x

diag

Specifies whether or not A is unit triangular:

  • diag = 'U' or 'u'. A is assumed to be unit triangular.

  • diag = 'N' or 'n'. A is not assumed to be unit triangular.

n

Specifies the order of the matrix A. n must be at least zero.

ap

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,n).

Before entry with uplo = 'U' or 'u', the leading (k+1) by n part of the array A must contain the upper triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row (k+1) of the array, the first super-diagonal starting at position 2 in row k, and so on. The top left k by k triangle of the array A is not referenced.

Before entry with uplo = 'L' or 'l', the leading (k+1) by n part of the array A must contain the lower triangular band part of the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row 1 of the array, the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right k by k triangle of the array A is not referenced.

Note that when diag = 'U' or 'u', the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least (1+(n-1)*abs(incx)). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the transformed vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_TPSV Procedures

This procedure solves one of the systems of equations

A*x = b

or

A'*x = b

where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular matrix, supplied in packed form.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_TPSV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   ap     IN      UTL_NLA_ARRAY_DBL,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_TPSV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   ap     IN      UTL_NLA_ARRAY_FLT,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-34 BLAS_TPSV Procedure Parameters

Parameter Description

uplo

Specifies whether the matrix is an upper or lower triangular matrix:

  • uplo = 'U' or 'u' : A is an upper triangular matrix.

  • uplo = 'L' or 'l' : A is a lower triangular matrix.

trans

Specifies the operation to be performed:

  • trans = 'N' or 'n' : A*x = b

  • trans = 'T' or 't' : A'*x = b

  • trans = 'C' or 'c' : A'*x = b

diag

Specifies whether or not A is unit triangular:

  • diag = 'U' or 'u' : A is assumed to be unit triangular.

  • diag = 'N' or 'n' : 'A is not assumed to be unit triangular.

n

Specifies the order of the matrix A. n must be at least zero.

ap

UTL_NLA_ARRAY_FLT/DBL of dimension at least

((n*(n+1))/2)

Before entry with uplo = 'U' or 'u', the array ap must contain the upper triangular matrix packed sequentially, column by column, so that ap(1) contains a(1,1), ap(2) and ap(3) contain a(1,2) and a(2,2) respectively, and so on.

Before entry with uplo = 'L' or 'l', the array ap must contain the lower triangular matrix packed sequentially, column by column, so that ap(1) contains a(1,1), ap(2) and ap(3) contain a(2,1) and a(3,1) respectively, and so on.

Note that when diag = 'U' or 'u', the diagonal elements of A are not referenced, but are assumed to be unity.

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

( 1 + ( n - 1 ) *abs( incx ) )

Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x.

incx

Specifies the increment for the elements of x. incx must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

Usage Notes

No test for singularity or near-singularity is included in this routine. Such tests must be performed before calling this routine.

BLAS_TRMM Procedures

This procedure performs one of the matrix-matrix operations

B := alpha*op( A )*B

or

B := alpha*B*op( A )

where alpha is a scalar, B is an m by n matrix, A is a unit, or non-unit, upper or lower triangular matrix and op(A) is one of

op( A ) = A

or

op( A ) = A'

See Also:

BLAS Level 3 (Matrix-Matrix Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_TRMM (
   side   IN      flag,
   uplo   IN      flag,
   transa IN      flag,
   diag   IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   b      IN OUT  UTL_NLA_ARRAY_DBL,
   ldb    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_TRMM (
   side   IN      flag,
   uplo   IN      flag,
   transa IN      flag,
   diag   IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   b      IN OUT  UTL_NLA_ARRAY_FLT,
   ldb    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-35 BLAS_TRMM Procedure Parameters

Parameter Description

side

Specifies whether the symmetric matrix A appears on the left or right in the operation:

  • side = 'L' or 'l' : B := alpha*op(A)*B

  • side = 'R' or 'r' : B := alpha*B*op(A)

uplo

Specifies whether the upper or lower triangular part of the array A is to be referenced:

  • uplo = 'U' or 'u' : A is an upper triangular matrix.

  • uplo = 'L' or 'l'' : A is a lower triangular matrix.

transa

Specifies the form of op(A) to be used in the matrix multiplication as follows:

  • transa = 'N' or 'n' : op(A) = A

  • transa = 'T' or 't' : op(A) = A'

  • transa ='C' or 'c' : op(A) = A'

diag

Specifies whether or not A is unit triangular:

  • diag = 'U' or 'u'. A is assumed to be unit triangular.

  • diag = 'N' or 'n'. A is not assumed to be unit triangular.

m

Specifies the number of rows of the B. m must be at least zero.

n

Specifies the number of columns of B. n must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha. When alpha is zero then A is not referenced and B need not be set before entry.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda,k) where k is m when side = 'L' or 'l', and is n when side = ''R' or 'r'.

Before entry with uplo = 'U' or 'u' , the leading k by k upper triangular part of the array A must contain the upper triangular matrix, and the strictly lower triangular part of A is not referenced.

Before entry with uplo = 'L' or 'l', the leading k byk lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced.

Note that when diag = ''U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity.

lda

Specifies the first dimension of a as declared in the calling (sub) program. When side = 'L' or 'l', lda must be at least max(1,m), otherwise lda must be at least max(1,n).

b

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (ldb,n).

Before entry, the leading m by n part of the array B must contain the matrix B, and on exit is overwritten by the transformed matrix.

ldb

Specifies the first dimension of b as declared in the calling (sub) program. ldb must be at least max(1,m).

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_TRMV Procedures

This procedure performs the matrix-vector operations

x := A*x

or

x := A'*x

where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular matrix.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_TRMV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_TRMV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-36 BLAS_TRMV Procedure Parameters

Parameter Description

uplo

Specifies whether the matrix is an upper or lower triangular matrix:

  • uplo = 'U' or 'u'. A is an upper triangular matrix.

  • uplo = 'L' or 'l'. A is a lower triangular matrix.

trans

Specifies the operation to be performed:

  • trans = 'N' or 'n'x := A*x

  • trans= 'T' or 't'x := A'*x

  • trans = 'C' or 'c'x := A'*x

diag

Specifies whether or not A is unit triangular:

  • diag = 'U' or 'u'. A is assumed to be unit triangular.

  • diag = 'N' or 'n'. A is not assumed to be unit triangular.

n

Specifies the order of the matrix A. n must be at least zero.  

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, n ).

Before entry with uplo = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular matrix and the strictly lower triangular part of A is not referenced.

Before entry with uplo = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced.

Note that when diag = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity

lda

Specifies the first dimension of a as declared in the calling (sub) program. lda must be at least max(1,n).

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least(1+(n-1)*as(incx)). Before entry, the incremented array X must contain the n element vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_TRSM Procedures

This procedure performs one of the matrix-matrix operations

op( A )*X = alpha*B

or

X*op( A ) = alpha*B

where alpha is a scalar, X and B are m by n matrices, A is a unit, or non-unit, upper or lower triangular matrix and op(A) is one of

op( A ) = A

or

op( A ) = A'

The matrix X is overwritten on B.

See Also:

BLAS Level 3 (Matrix-Matrix Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_TRSM (
   side   IN      flag,
   uplo   IN      flag,
   transa IN      flag,
   diag   IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_DOUBLE,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   b      IN OUT  UTL_NLA_ARRAY_DBL,
   ldb    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_TRSM (
   side   IN      flag,
   uplo   IN      flag,
   transa IN      flag,
   diag   IN      flag,
   m      IN      POSITIVEN,
   n      IN      POSITIVEN,
   alpha  IN      SCALAR_FLOAT,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   b      IN OUT  UTL_NLA_ARRAY_FLT,
   ldb    IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-37 BLAS_TRSM Procedure Parameters

Parameter Description

side

Specifies whether the symmetric matrix A appears on the left or right in the operation:

  • side = 'L' or 'l' : op(A)*X = alpha*B

  • side = 'R' or 'r' : X*op(A) = alpha*B

uplo

Specifies whether the upper or lower triangular part of the array A is to be referenced:

  • uplo = 'U' or 'u' : A is an upper triangular matrix.

  • uplo = 'L' or 'l' : A is a lower triangular matrix.

transa

Specifies the form of op(A) to be used in the matrix multiplication as follows:

  • transa = 'N' or 'n' : op(A) = A

  • transa ='T' or 't' : op(A) = A'

  • transa = 'C' or 'c' : op(A) = A'

diag

Specifies whether or not A is unit triangular:

  • diag = 'U' or 'u'. A is assumed to be unit triangular.

  • diag = 'N' or 'n'. A is not assumed to be unit triangular.

m

Specifies the number of rows of the B. m must be at least zero.

n

Specifies the number of columns of B. n must be at least zero.

alpha

SCALAR_FLOAT/DOUBLE. Specifies the scalar alpha. When alpha is zero then A is not referenced and B need not be set before entry.

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, k) where k is m when side = 'L' or 'l', and is n when side = 'R' or 'r'.

Before entry with uplo = 'U' or 'u', the leading k by k upper triangular part of the array A must contain the upper triangular matrix, and the strictly lower triangular part of A is not referenced.

Before entry with uplo = 'L' or 'l', the leading k by k lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced.

Note that when diag = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity.

lda

Specifies the first dimension of a as declared in the calling (sub) program. When side = 'L' or 'l', lda must be at least max(1, m), otherwise lda must be at least max (1, n).

b

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (ldb, n).

Before entry, the leading m by n part of the array B must contain the matrix B, and on exit is overwritten by the solution matrix X.

ldb

Specifies the first dimension of b as declared in the calling (sub) program. ldb must be at least max(1, m).

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

BLAS_TRSV Procedures

This procedure solves one of the systems of equations

A*x = b

or

A'*x = b

where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular matrix.

See Also:

BLAS Level 2 (Matrix-Vector Operations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.BLAS_TRSV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   a      IN      UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_DBL,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.BLAS_TRSV (
   uplo   IN      flag,
   trans  IN      flag,
   diag   IN      flag,
   n      IN      POSITIVEN,
   a      IN      UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   x      IN OUT  UTL_NLA_ARRAY_FLT,
   incx   IN      POSITIVEN,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-38 BLAS_TRSV Procedure Parameters

Parameter Description

uplo

Specifies whether the matrix is an upper or lower triangular matrix:

  • uplo = 'U' or 'u'. A is an upper triangular matrix.

  • uplo = 'L' or 'l'. A is a lower triangular matrix.

trans

Specifies the operation to be performed:

  • trans = 'N' or 'n'A*x = b

  • trans = 'T' or 't'A'*x = b

  • trans = 'C' or 'c'A'*x = b

diag

Specifies whether or not A is unit triangular:

  • diag = 'U' or 'u'. A is assumed to be unit triangular.

  • diag = 'N' or 'n'. A is not assumed to be unit triangular.

n

Specifies the order of the matrix A. n must be at least zero.  

a

UTL_NLA_ARRAY_FLT/DBL of DIMENSION (lda, n).

Before entry with uplo = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular matrix and the strictly lower triangular part of A is not referenced.

Before entry with uplo = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced.

Note that when diag = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity.

lda

Specifies the first dimension of A as declared in the calling (sub) program. lda must be at least max(1, n).

x

UTL_NLA_ARRAY_FLT/DBL of dimension at least

(1 + (n - 1) * abs (incx))

Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x.

incx

Specifies the increment for the elements of x. Must not be zero.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

Usage Notes

No test for singularity or near-singularity is included in this routine. Such tests must be performed before calling this routine.

LAPACK_GBSV Procedures

This procedure computes the solution to a real system of linear equations

a * x = b

where a is a band matrix of order n with kl sub diagonals and ku superdiagonals, and x and b are n by nrhs matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as

a = L * U

where L is a product of permutation and unit lower triangular matrices with kl sub diagonals, and U is upper triangular with kl+ku superdiagonals. The factored form of a is then used to solve the system of equations

a * x = b

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_GBSV (
   n      IN      POSITIVEN,
   kl     IN      NATURALN,
   ku     IN      NATURALN,
   nrhs   IN      POSITIVEN,
   ab     IN OUT  UTL_NLA_ARRAY_DBL,
   ldab   IN      POSITIVEN,
   ipiv   IN OUT  UTL_NLA_ARRAY_INT,
   b      IN OUT  UTL_NLA_ARRAY_DBL,
   ldb    IN      POSITIVEN,
   info   OUT  INTEGER,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_GBSV (
   n      IN      POSITIVEN,
   kl     IN      NATURALN,
   ku     IN      NATURALN,
   nrhs   IN      POSITIVEN,
   ab     IN OUT  UTL_NLA_ARRAY_FLT,
   ldab   IN      POSITIVEN,
   ipiv   IN OUT  UTL_NLA_ARRAY_INT,
   b      IN OUT  UTL_NLA_ARRAY_FLT,
   ldb    IN      POSITIVEN,
   info   OUT  INTEGER,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-39 LAPACK_GBSV Procedure Parameters

Parameter Description

n

The number of linear equations, equivalent to the order of the matrixa .n >= 0

kl

The number of sub diagonals within the band of a. kl >= 0.

ku

The number of superdiagonals within the band of a . ku >= 0.

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

ab

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldab, n).

On entry, the matrix a in band storage, in rows kl+1 to 2*kl+ku+1; rows 1 to kl of the array need not be set. The j-th column of A is stored in the j-th column of the array ab:

ab(kl+ku+1+i-j,j) = a(i,j) for max(1,j-ku)<=i<=min(n,j+kl)

On exit, details of the factorization: U is stored as an upper triangular band matrix with kl+ku superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows:

kl+ku+2 to 2*kl+ku+1

ldab

The leading dimension of the array ab.

ldab >= 2*kl+ku+1

ipiv

INTEGER array, DIMENSION (n).

The pivot indices that define the permutation matrix P; row iof the matrix was interchanged with row ipiv(i).

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the n by nrhs matrix of right hand side matrix b.

On exit, if info = 0 , the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max(1,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i , the i-th argument had an illegal value

  • > 0 : if info = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_GEES Procedures

This procedure computes for an n by n real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).

A matrix is in real Schur form if it is upper quasi-triangular with 1 by 1 and 2 by 2 blocks. 2 by 2 blocks will be standardized in the form

[ a b ]

[ c a ]

where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_GEES (
   jobvs   IN       flag,
   n       IN       POSITIVEN,
   a       IN  OUT  UTL_NLA_ARRAY_DBL,
   lda     IN       POSITIVEN,
   wr      IN OUT   UTL_NLA_ARRAY_DBL,
   wi      IN OUT   UTL_NLA_ARRAY_DBL,
   vs      IN OUT   UTL_NLA_ARRAY_DBL,
   ldvs    IN       POSITIVEN,
   info    OUT      INTEGER,
   pack    IN       flag DEFAULT 'C');

UTL_NLA.LAPACK_GEES (
   jobvs   IN       flag,
   n       IN       POSITIVEN,
   a       IN OUT   UTL_NLA_ARRAY_FLT,
   lda     IN       POSITIVEN,
   wr      IN       OUT UTL_NLA_ARRAY_FLT,
   wi      IN       OUT UTL_NLA_ARRAY_FLT,
   vs      IN OUT   UTL_NLA_ARRAY_FLT,
   ldvs    IN       POSITIVEN,
   info    OUT      integer,
   pack    IN       flag DEFAULT 'C');

Parameters

Table 232-40 LAPACK_GEES Procedure Parameters

Parameter Description

jobz

  • 'N': Schur vectors are not computed.

  • 'V': Schur vectors are computed.

n

The order of the matrix a. N >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

  • On entry, the n by n matrix A.

  • On exit, A has been overwritten by its real Schur form T.

lda

The leading dimension of the array a. lda >= max(1,n).

wr

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

wr and wi contain the real and imaginary parts respectively of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form T. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

wi

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldz, n).

wr and wi contain the real and imaginary parts respectively of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form T. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

vs

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

  • If jobvs = 'V', vs contains the orthogonal matrix Z of Schur vectors.

  • If jobvs = 'N', vs is not referenced.

ldvs

The leading dimension of the array vs. VS. ldvs >= 1. If jobvs = 'V', ldvs >= N

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, and i is <= N: the QR algorithm failed to compute all the eigenvalues. Elements 1:ILO-1 and i+1:N of wr and wi contain those eigenvalues which have converged. If jobvs = 'V', vs contains the matrix which reduces A to its partially converged Schur form.

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_GELS Procedures

This procedure solves overdetermined or underdetermined real linear systems involving an m by n matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank.

The following options are provided:

  1. If TRANS = 'N' and m >= n: find the least squares solution of an overdetermined system, that is, solve the least squares problem.

    minimize || B - A*X ||

  2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B.

  3. If TRANS = 'T' and m >= n: find the minimum norm solution of an undetermined system A**T * X = B.

  4. If TRANS = 'T' and m < n: find the least squares solution of an overdetermined system, that is, solve the least squares problem minimize || B - A**T * X ||.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_GELS (
   trans   IN       flag,
   m       IN       POSITIVEN,
   n       IN       POSITIVEN,
   nrhs    IN       POSITIVEN,
   a       IN OUT   UTL_NLA_ARRAY_DBL,
   lda     IN       POSITIVEN,
   b       IN OUT   UTL_NLA_ARRAY_DBL,
   ldb     IN       POSITIVEN,
   info    OUT      INTEGER,
   pack    IN       flag DEFAULT 'C');

UTL_NLA.LAPACK_GELS (
  trans    IN       flag,
  m        IN       POSITIVEN,
  n        IN       POSITIVEN,
  nrhs     IN       POSITIVEN,
  a        IN OUT   UTL_NLA_ARRAY_FLT,
  lda      IN       POSITIVEN,
  b        IN OUT   UTL_NLA_ARRAY_FLT,
  ldb      IN       POSITIVEN,
  info     OUT      INTEGER,
  pack     IN       flag DEFAULT 'C');

Parameters

Table 232-41 LAPACK_GELS Procedure Parameters

Parameter Description

trans

  • CHARACTER = 'N': The linear system involves A.

  • CHARACTER = 'T': The linear system involves A**T .

m

The number of rows of the matrix a. M >= 0.

n

The number of columns of the matrix a. N >= 0.

nrhs

The number of right-hand sides, which is the number of columns of the matrix band x.nrhs >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

On entry, the matrix b of right hand side vectors, stored columnwise; b is m by nrhs if TRANS = 'N', or n by nrhsif trans = 'T'.

On exit, if m >= n, a is overwritten by details of its QR factorization as returned by SGEQRF. If m < n, A is overwritten by details of its LQ factorization as returned by SGELQF.

lda

The leading dimension of the array A. lda >= max(1,m).

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the matrix b of right hand side vectors, stored columnwise. b is m bynrhs if trans = 'n', or n by nrhs if trans = 'T'.

On exit, b is overwritten by the solution vectors, stored columnwise:

  • If trans = 'n' and m >= n, rows 1 to n of b contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements n+1 to m in that column.

  • If trans = 'n' and m < n, rows 1 to n of b contain the minimum norm solution vectors.

  • If trans = 'T' and m >= n, rows 1 to m of b contain the minimum norm solution vectors.

  • If trans = 'T' and m < n, rows 1 to m of b contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements m+1 to n in that column.

ldb

The leading dimension of the array b.

ldb >= max(1,m,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_GESDD Procedures

This procedures computes the singular value decomposition (SVD) of a real m by n matrix A, optionally computing the left and right singular vectors. If singular vectors are desired, it uses a divide-and-conquer algorithm that makes mild assumptions about floating point arithmetic.

The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an m by n matrix which is zero except for its min(m,n) diagonal elements, U is anm by m orthogonal matrix, and V is ann by n orthogonal matrix. The diagonal elements of SIGMA are the singular values of A, they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.

Note that the routine returns V**T, not V.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_GESDD (
   jobz   IN       flag,
   m      IN       POSITIVEN,
   n      IN       POSITIVEN,
   a      IN OUT   UTL_NLA_ARRAY_DBL,
   lda    IN       POSITIVEN,
   s      IN OUT   UTL_NLA_ARRAY_DBL,
   u      IN OUT   UTL_NLA_ARRAY_DBL,
   ldu    IN       POSITIVEN,
   vt     IN OUT   UTL_NLA_ARRAY_DBL,
   ldvt   IN       POSITIVEN,
   info   OUT      INTEGER,
   pack   IN       flag DEFAULT 'C');

UTL_NLA.LAPACK_GESDD (
   jobz   IN       flag,
   m      IN       POSITIVEN,
   n      IN       POSITIVEN,
   a      IN OUT   UTL_NLA_ARRAY_FLT,
   lda    IN       POSITIVEN,
   s      IN OUT   UTL_NLA_ARRAY_FLT,
   u      IN OUT   UTL_NLA_ARRAY_FLT,
   ldu    IN       POSITIVEN,
   vt     IN OUT   UTL_NLA_ARRAY_FLT,
   ldvt   IN       POSITIVEN,
   info   OUT      INTEGER,
   pack   IN       flag DEFAULT 'C');

Parameters

Table 232-42 LAPACK_GESDD Procedure Parameters

Parameter Description

jobz

Specifies options for computing all or part of the matrix U:

  • 'A': All m columns of u and all n rows of V**T are returned in arrays u and vt.

  • 'S': The first min(m,n) columns of u and the first min(m,n) rows of V**T are returned in the arrays u and vt.

  • 'O': The first min(m,n) columns of u (the left singular vectors) are overwritten on the array a. jobu and jobvt cannot both be 'O'

  • 'N': No columns of u (no left singular vectors) are computed.

m

The order of the matrix a. m >= 0.

n

The order of the matrix a. n >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

On entry, the n by n matrix A.

On exit:

  • If jobz = 'O', a is overwritten with the first min(m,n) columns of u (the left singular vectors, stored columnwise).

  • If m >= n, a is overwritten with the first m rows of V**T (the right singular vectors, stored rowwise).

  • If jobz .ne. 'O', the contents of a are destroyed.

lda

The leading dimension of the array a. lda >= max(1,m).

s

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (min(m,n).

The singular values of a, sorted so that S(i) >= S(i+1).

u

UTL_NLA_ARRAY_FLT/DBL. ucol = m if jobz = 'A' or jobz = 'O' and m < n; ucol = min(m,n) if jobz = 'S'.

  • If jobz = 'A' or jobz = 'O' and m < n, u contains the m by m orthogonal matrix u.

  • If jobz = 'S', u contains the first min(m,n) columns of u (the left singular vectors, stored columnwise).

  • If jobz = 'O' and m >= n, or jobz = 'n', u is not referenced.

ldu

The leading dimension of the array U. ldu >= 1. If jobz = 'S' or 'A',or jobz = 'O' and m < n, ldu >= m.

vt

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldvt, n).

  • If jobz = 'A' or jobz = 'O' and m >= n, vt contains the n by n orthogonal matrix V**T.

  • If jobz = 'S', vt contains the first min(m,n) rows of V**T (the right singular vectors, stored rowwise).

  • If jobz = 'O' and m < n, or jobz = 'N', vt is not referenced.

ldvt

The leading dimension of the array vt. ldvt >= 1.

  • If jobz = 'A', or jobz = 'O' and m >= n, ldvt >= n.

  • If jobz = 'S', ldvt >= min(m,n).

info

  • = 0 : successful exit

  • < 0 : If info = -i, the i-th argument had an illegal value

  • > 0 : SBDSDC did not converge, updating process failed.

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_GESV Procedures

This procedure computes the solution to a real system of linear equations

a * x = b

where a is an n by n matrix and x and b are n by nrhs matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as

a = P * L * U

where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of a is then used to solve the system of equations

a * x = b

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_GESV (
   n      IN      POSITIVEN,
   nrhs   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_DBL,
   lda    IN      POSITIVEN,
   ipiv   IN OUT  UTL_NLA_ARRAY_INT,
   b      IN OUT  UTL_NLA_ARRAY_DBL,
   ldb    IN      POSITIVEN,
   info      OUT  INTEGER,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_GESV (
   n      IN      POSITIVEN,
   nrhs   IN      POSITIVEN,
   a      IN OUT  UTL_NLA_ARRAY_FLT,
   lda    IN      POSITIVEN,
   ipiv   IN OUT  UTL_NLA_ARRAY_INT,
   b      IN OUT  UTL_NLA_ARRAY_FLT,
   ldb    IN      POSITIVEN,
   info      OUT  INTEGER,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-43 LAPACK_GESV Procedure Parameters

Parameter Description

n

The number of linear equations, equivalent to the order of the matrix a. n >= 0

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

On entry, the n by n coefficient matrix a.

On exit, the factors L and U from the factorization a = P*L*U; the unit diagonal elements of L are not stored.

lda

The leading dimension of the array a.

lda >= max(1,n)

ipiv

INTEGER array, DIMENSION (n).

The pivot indices that define the permutation matrix P; row iof the matrix was interchanged with row ipiv(i).

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the n by nrhs matrix of right hand side matrix b.

On exit, if info = 0 , the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max(1,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i , the i-th argument had an illegal value

  • > 0 : if info = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_GESVD Procedures

This procedures computes the singular value decomposition (SVD) of a real m by n matrix A, optionally computing the left and/or right singular vectors. The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an m by n matrix which is zero except for its min(m,n) diagonal elements, U is anm by m orthogonal matrix, and V is ann by n orthogonal matrix. The diagonal elements of SIGMA are the singular values of A, they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.

Note that the routine returns V**T, not V.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_GESVD (
   jobu   IN       flag,
   jobvt  IN       flag,
   m      IN       POSITIVEN,
   n      IN       POSITIVEN,
   a      IN OUT   UTL_NLA_ARRAY_DBL,
   lda    IN       POSITIVEN,
   s      IN OUT   UTL_NLA_ARRAY_DBL,
   u      IN OUT   UTL_NLA_ARRAY_DBL,
   ldu    IN       POSITIVEN,
   vt     IN OUT   UTL_NLA_ARRAY_DBL,
   ldvt   IN       POSITIVEN,
   info   OUT      INTEGER,
   pack   IN       flag DEFAULT 'C');

UTL_NLA.LAPACK_GESVD (
   jobu   IN       flag,
   jobvt  IN       flag,
   m      IN       POSITIVEN,
   n      IN       POSITIVEN,
   a      IN OUT   UTL_NLA_ARRAY_FLT,
   lda    IN       POSITIVEN,
   s      IN OUT   UTL_NLA_ARRAY_FLT,
   u      IN OUT   UTL_NLA_ARRAY_FLT,
   ldu    IN       POSITIVEN,
   vt     IN OUT   UTL_NLA_ARRAY_FLT,
   ldvt   IN       POSITIVEN,
   info   OUT      INTEGER,
   pack   IN       flag DEFAULT 'C');

Parameters

Table 232-44 LAPACK_GESVD Procedure Parameters

Parameter Description

jobu

Specifies options for computing all or part of the matrix U:

  • 'A': All m columns of U are returned in array U.

  • 'S': The first min(m,n) columns of U (the left singular vectors) are returned in the array U.

  • 'O': The first min(m,n) columns of U (the left singular vectors) are overwritten on the array a. jobu and jobvt cannot both be 'O

  • 'N': No columns of U (no left singular vectors) are computed.

jobvt

Specifies options for computing all or part of the matrix V**T:

  • 'A': All n rows of V**T are returned in the array vt.

  • 'S': The first min(m,n) rows of V**T (the right singular vectors) are returned in the array vt.

  • 'O': The first min(m,n) rows of V**T (the right singular vectors) are overwritten on the array a. jobvt and jobu cannot both be 'O'.

  • 'N': No rows of V**T (no right singular vectors) are computed.

m

The order of the matrix a. M >= 0.

n

The order of the matrix a. N >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

On entry, the n by n matrix A.

On exit:

  • If jobu = 'O', A is overwritten with the first min(m,n) columns of U (the left singular vectors, stored columnwise);

  • If jobvt = 'O', A is overwritten with the first min(m,n) rows of V**T (the right singular vectors, stored rowwise);

  • If jobu.ne.'O' and jobvt.ne.'O', the contents of A are destroyed.

lda

The leading dimension of the array a. lda >= max(1,n).

s

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (min(m,n).

The singular values of A, sorted so that S(i) >= S(i+1).

u

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldu,ucol).(ldu,m)if jobu = 'A' or (ldu,min(m,n)) if jobu = 'S'.

  • If jobu = 'A', U contains the m by m orthogonal matrix U.

  • If jobu = 'S', U contains the first min(m,n) columns of U (the left singular vectors, stored columnwise).

  • If jobu = 'N' or 'O', U is not referenced.

ldu

The leading dimension of the array U. ldu >= 1. If jobu = 'S' or 'a', ldu >= m.

vt

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldvt, n).

  • If jobvt = 'A', vt contains the n by n orthogonal matrix V**T.

  • If jobvt = 'S', vt contains the first min(m,n) rows of V**T (the right singular vectors, stored rowwise).

  • If jobvt = 'N' or 'O', vt is not referenced.

ldvt

The leading dimension of the array vt. ldvt >= 1.

  • If jobvt = 'A', ldvt >= n.

  • If jobvt = 'S', ldvt >= min(m,n).

info

  • = 0 : successful exit

  • < 0 : If info = -i, the i-th argument had an illegal value

  • > 0 : If SBDSQR did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form B did not converge to zero.

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_GEEV Procedures

This procedures computes for an n by n real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

  • The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue.

  • The left eigenvector u(j) of A satisfies u(j)**H * A = lambda(j) * u(j)**H where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_GEEV (
   jobvl   IN      flag,
   jobvr   IN      flag,
   n       IN      POSITIVEN,
   a       IN OUT  UTL_NLA_ARRAY_DBL,
   lda     IN      POSITIVEN,
   wr      IN OUT  UTL_NLA_ARRAY_DBL,
   wi      IN OUT  UTL_NLA_ARRAY_DBL,
   vl      IN OUT  UTL_NLA_ARRAY_DBL,
   ldvl    IN      POSITIVEN,
   vr      IN OUT  UTL_NLA_ARRAY_DBL,
   ldvr    IN      POSITIVEN,
   info    OUT     INTEGER,
   pack    IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_GEEV (
   jobvl   IN      flag,
   jobvr   IN      flag,
   n       IN      POSITIVEN,
   a       IN OUT  UTL_NLA_ARRAY_FLT,
   lda     IN      POSITIVEN,
   wr      IN OUT  UTL_NLA_ARRAY_FLT,
   wi      IN OUT  UTL_NLA_ARRAY_FLT,
   vl      IN OUT  UTL_NLA_ARRAY_FLT,
   ldvl    IN      POSITIVEN,
   vr      IN OUT  UTL_NLA_ARRAY_FLT,
   ldvr    IN      POSITIVEN,
   info    OUT     INTEGER,
   pack    IN      flag DEFAULT 'C');

Parameters

Table 232-45 LAPACK_GEEV Procedure Parameters

Parameter Description

jobvl

  • 'N': Left eigenvectors of A are not computed.

  • 'V': Left eigenvectors of A are computed.

jobvr

  • 'N': Right eigenvectors of A are not computed.

  • 'V': Right eigenvectors of A are computed.

n

The order of the matrix a. N >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

  • On entry, the n by n matrix A.

  • On exit, A has been overwritten.

lda

The leading dimension of the array a. lda >= max(1,n).

wr

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

wr and wi contain the real and imaginary parts respectively of the computed eigenvalues. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

wi

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldz, n).

wr and wi contain the real and imaginary parts respectively of the computed eigenvalues. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

vl

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

  • If jobvl = 'V', the left eigenvectors u(j) are stored one after another in the columns of vl, in the same order as their eigenvalues.

  • If jobvs = 'N', vl is not referenced.

  • If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of vl.

  • If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and u(j+1) = VL(:,j) - i*VL(:,j+1).

ldv1

The leading dimension of the array vl. ldvl >= 1. If jobvl = 'v', ldvl >= n.

vr

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldvr, n).

  • If jobvr = 'V', the right eigenvectors v(j) are stored one after another in the columns of vr, in the same order as their eigenvalues..

  • If jobvr = 'N', vr is not referenced.

  • If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of vr.

  • If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and v(j+1) = VR(:,j) - i*VR(:,j+1).

ldvr

The leading dimension of the array vr. vr.ldvr >= 1. If jobvr = 'V', ldvr >= N

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, and i is <= N: the QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed. Elements i+1:N of wr and wi contain eigenvalues which have converged..

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_GTSV Procedures

This procedure solves the equation

a * x = b

where a is an n by n tridiagonal matrix, by Gaussian elimination with partial pivoting.

Note that the equation a'*x = b may be solved by interchanging the order of the arguments du and dl.

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_GTSV (
   n      IN      POSITIVEN,
   nrhs   IN      POSITIVEN,
   dl     IN OUT  UTL_NLA_ARRAY_DBL,
   d      IN OUT  UTL_NLA_ARRAY_DBL,
   du     IN OUT  UTL_NLA_ARRAY_DBL,
   b      IN OUT  UTL_NLA_ARRAY_DBL,
   ldb    IN      POSITIVEN,
   info      OUT  INTEGER,
   pack   IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_GTSV (
   n      IN      POSITIVEN,
   nrhs   IN      POSITIVEN,
   dl     IN OUT  UTL_NLA_ARRAY_FLT,
   d      IN OUT  UTL_NLA_ARRAY_FLT,
   du     IN OUT  UTL_NLA_ARRAY_FLT,
   b      IN OUT  UTL_NLA_ARRAY_FLT,
   ldb    IN      POSITIVEN,
   info      OUT  INTEGER,
   pack   IN      flag DEFAULT 'C');

Parameters

Table 232-46 LAPACK_GTSV Procedure Parameters

Parameter Description

n

The order of the matrix a .n >= 0

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

dl

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n-1).

On entry, dl must contain the (n-1) sub-diagonal elements of a.

On exit, dl is overwritten by the (n-2) elements of the second super-diagonal of the upper triangular matrix U from the LU factorization of a, in dl(1), ..., dl(n-2).

d

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

On entry, d must contain the diagonal elements of a.

On exit, d is overwritten by the n diagonal elements of U.

du

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n-1).

On entry, du must contain the (n-1)super-diagonal elements of a.

On exit, du is overwritten by the (n-1) elements of the first super-diagonal of U.

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (LDB, nrhs).

On entry, the n by nrhs matrix of right hand side matrix b.

On exit, if info = 0, the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max (1, n)

info

  • = 0 : successful exit

  • < 0 : if info = -i , the i-th argument had an illegal value

  • > 0 : if info = i, U(i,i) is exactly zero, and the solution has not been computed. The factorization has not been completed unless i = n.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_PBSV Procedures

This procedure computes the solution to a real system of linear equations

a * x = b

where a is an n by n symmetric positive definite band matrix and x and b are n by nrhs matrices.

The Cholesky decomposition is used to factor A as

A = U**T*U if UPLO ='U'

or

A = L * L**T if UPLO = 'L'

where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B.

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_PBSV (
uplo       IN      flag,
n          IN      POSITIVEN,
kd         IN      NATURALN,
nrhs       IN      POSITIVEN,
ab         IN OUT  UTL_NLA_ARRAY_DBL,
ldab       IN      POSITIVEN,
b          IN OUT  UTL_NLA_ARRAY_DBL,
ldb        IN      POSITIVEN,
info       OUT     INTEGER,
pack       IN      flag DEFAULT 'C');
    

UTL_NLA.LAPACK_PBSV (
uplo      IN      flag,
n         IN      POSITIVEN,
kd        IN      NATURALN,
nrhs      IN      POSITIVEN,
ab        IN OUT  UTL_NLA_ARRAY_FLT,
ldab      IN      POSITIVEN,
b         IN OUT  UTL_NLA_ARRAY_FLT,
ldb       IN      POSITIVEN,
info      OUT     INTEGER,
pack      IN      flag DEFAULT 'C');

Parameters

Table 232-47 LAPACK_PBSV Procedure Parameters

Parameter Description

uplo

  • uplo = 'U'. Upper triangular of A is stored.

  • uplo = 'L'. Lower triangular of A is stored.

n

The number of linear equations, that is, the order of the matrix a .n >= 0

kd

The number of superdiagonals of the matrix A if uplo = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

ab

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldab, n).

On entry, the upper or lower triangle of the symmetric band matrix a, stored in the first kd+1 rows of the array. The j-th column of a is stored in the j-th column of the array ab is as follows:

  • if uplo = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;

  • if uplo = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD)

.See below for further details.On exit, if info = 0, the triangular factor U or L from theCholesky factorization A = U**T*U or A = L*L**T of the bandmatrix A, in the same storage format as a.

ldab

The leading dimension of the array ab.

ldb >= kd+1

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the n by nrhs matrix of right hand side matrix b.

On exit, if info = 0 , the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max(1,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i , the i-th argument had an illegal value

  • > 0 : if info = i, the leading minor of order i of a is not positive definite, so the factorization could not be completed, and the solution has not been computed.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_POSV Procedures

This procedure computes the solution to a real system of linear equations

a * x = b

where a is an n by n symmetric positive definite matrix and x and b are n by nrhs matrices.

The Cholesky decomposition is used to factor A as

A = U**T* U if uplo = 'U'

or

A = L * L**T if UPLO = 'L'

where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B.

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_POSV (
   uplo      IN      flag,
   n         IN      POSITIVEN,
   nrhs      IN      POSITIVEN,
   a         IN OUT  UTL_NLA_ARRAY_DBL,
   lda       IN      POSITIVEN,
   b         IN OUT  UTL_NLA_ARRAY_DBL,
   ldb       IN      POSITIVEN,
   info      OUT     INTEGER,
   pack      IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_POSV (
   uplo      IN      flag,
   n         IN      POSITIVEN,
   nrhs      IN      POSITIVEN,
   a         IN OUT  UTL_NLA_ARRAY_FLT,
   lda       IN      POSITIVEN,
   b         IN OUT  UTL_NLA_ARRAY_FLT,
   ldb       IN      POSITIVEN,
   info      OUT     INTEGER,
   pack      IN      flag DEFAULT 'C');

Parameters

Table 232-48 LAPACK_POSV Procedure Parameters

Parameter Description

uplo

  • uplo = 'U'. Upper triangular of A is stored.

  • uplo = 'L'. Lower triangular of A is stored.

n

The number of linear equations, that is, the order of the matrix a .n >= 0

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

If uplo = 'U', the leading NRHS n by n upper triangular part of a contains the upper NRHS triangular part of the matrix A, and the strictly lower NRHS triangular part of A is not referenced.

If uplo = ''L', then rhs leading n by n lower triangular part of a contains the lower nrhs triangular part of the matrix a, and the strictly upper nrhs triangular part of a is not referenced.

On exit, if info = 0, the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.

lda

The leading dimension of the array a.

lda >= max (1, n)

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the n by nrhs matrix of right hand side matrix b.

On exit, if info = 0 , the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max(1,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i , the i-th argument had an illegal value

  • > 0 : if info = i, the leading minor of order i of a is not positive definite, so the factorization could not be completed, and the solution has not been computed.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_PPSV Procedures

This procedure computes the solution to a real system of linear equations

a * x = b

where a is an n by n symmetric positive definite matrix stored in packed format and x and b are n by nrhs matrices.

The Cholesky decomposition is used to factor A as

A = U**T* U if UPLO = 'U'

or

A = L * L**T if UPLO = 'L'

where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B.

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_PPSV (
   uplo      IN      flag,
   n         IN      POSITIVEN,
   nrhs      IN      POSITIVEN,
   ap        IN OUT  UTL_NLA_ARRAY_DBL,
   b         IN OUT  UTL_NLA_ARRAY_DBL,
   ldb       IN      POSITIVEN,
   info      OUT     INTEGER,
   pack      IN      flag DEFAULT 'C');
    

UTL_NLA.LAPACK_PPSV (
   uplo      IN      flag,
   n         IN      POSITIVEN,
   nrhs      IN      POSITIVEN,
   ap        IN OUT  UTL_NLA_ARRAY_FLT,
   b         IN OUT  UTL_NLA_ARRAY_FLT,
   ldb       IN      POSITIVEN,
   info      OUT     INTEGER,
   pack      IN      flag DEFAULT 'C');

Parameters

Table 232-49 LAPACK_PPSV Procedure Parameters

Parameter Description

uplo

  • uplo = 'U' . Upper triangular of A is stored.

  • uplo = 'L'. Lower triangular of A is stored.

n

The number of linear equations, that is, the order of the matrix a .n >= 0

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

ap

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n*(n+1)/2).

On entry, the upper or lower triangle of the symmetric matrix a, packed columnwise in a linear array. The j-th column of a is stored in the array ap as follows:

If uplo = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

If uplo = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n;

On exit, if info = 0, the factor U or 'L' from the Cholesky factorization A = U**T*U or A = L*L**T in the same storage format as A.

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the n by nrhs matrix of right hand side matrix b.

On exit, if info = 0 , the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max(1,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i , the i-th argument had an illegal value

  • > 0 : if info = i, the leading minor of order i of a is not positive definite, so the factorization could not be completed, and the solution has not been computed.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_PTSV Procedures

This procedure computes the solution to a real system of linear equations

a * x = b

where a is an n by n symmetric positive definite tridiagonal matrix, and x and b are n by nrhs matrices.

a is factored as A = L*D*L**T, and the factored form of a is then used to solve the system of equations.

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_PTSV (
   n      IN       POSITIVEN,
   nrhs   IN       POSITIVEN,
   d      IN OUT   UTL_NLA_ARRAY_DBL,
   e      IN OUT   UTL_NLA_ARRAY_DBL,
   b      IN OUT   UTL_NLA_ARRAY_DBL,
   ldb    IN       POSITIVEN,
   info   OUT      INTEGER,
   pack   IN       flag  DEFAULT 'C');
 

UTL_NLA.LAPACK_PTSV (
   n      IN       POSITIVEN,
   nrhs   IN       POSITIVEN,
   d      IN OUT   UTL_NLA_ARRAY_FLT,
   e      IN OUT   UTL_NLA_ARRAY_FLT,
   b      IN OUT   UTL_NLA_ARRAY_FLT,
   ldb    IN       POSITIVEN,
   info   OUT      INTEGER,
   pack   IN       flag DEFAULT 'C');

Parameters

Table 232-50 LAPACK_PTSV Procedure Parameters

Parameter Description

n

The order of the matrix a. N >= 0.

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

d

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

On entry, the n diagonal elements of the tridiagonal matrix a.

On exit, the n diagonal elements of the diagonal matrix d from the factorization A = L*D*L**T.

e

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n-1).

On entry, the (n-1) subdiagonal elements of the tridiagonal matrix a.

On exit, the (n-1) diagonal elements of the unit bidiagonal factor L from the factorization A = L*D*L**T of a.(e can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**T*D*U factorization of a)

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the n by nrhs matrix of right hand side matrix b.

On exit, if info = 0 , the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max(1,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i , the i-th argument had an illegal value

  • > 0 : if info = i, the leading minor of order i of a is not positive definite, so the factorization could not be completed, and the solution has not been computed.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_SBEV Procedures

This procedure computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_SBEV (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   kd       IN      NATURALN,
   ab       IN OUT  UTL_NLA_ARRAY_DBL,
   ldab     IN      POSITIVEN,
   w        IN OUT  UTL_NLA_ARRAY_DBL,
   z        IN OUT  UTL_NLA_ARRAY_DBL,
   ldz       IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_SBEV (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   kd       IN      NATURALN,
   ab       IN OUT  UTL_NLA_ARRAY_FLT,
   ldab     IN      POSITIVEN,
   w        IN OUT  UTL_NLA_ARRAY_FLT,
   z        IN OUT  UTL_NLA_ARRAY_FLT,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

Parameters

Table 232-51 LAPACK_SBEV Procedure Parameters

Parameter Description

jobz

  • 'N': Compute eigenvalues only.

  • 'V': Compute eigenvalues and eigenvectors.

uplo

  • 'U': Upper triangle of A is stored.

  • 'L': Lower triangle of A is stored.

n

The order of the matrix a. N >= 0.

kd

The number of superdiagonals of the matrix A if uplo = 'U', or the number of subdiagonals if uplo = 'L'. kd >= 0.

ab

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldab, n).

On entry, the upper or lower triangle of the symmetric band matrix A stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array ab:

  • If uplo = 'U', ab(kd+1+i-j,j) = a(i,j) for max(1,j-kd)<=i<=j.

  • If uplo = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, ab is overwritten by values generated during the reduction to tridiagonal form:

  • If uplo = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T are returned in rows kd and kd+1 of ab.

  • If uplo = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of ab.

ldab

The leading dimension of the array ab. ldab >= kd + 1.

w

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

If info = 0, the eigenvalues in ascending order.

z

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

  • If jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors of the matrix A, with the i-th column of z holding the eigenvector associated with w(i).

  • If jobz = 'N', then z is not referenced.

ldz

The leading dimension of the array z. ldz >= 1, and if jobz = 'v', ldz >= max(1,n).

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_SBEVD Procedures

This procedure computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_SBEVD (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   kd       IN      NATURALN,
   ab       IN OUT  UTL_NLA_ARRAY_DBL,
   ldab     IN      POSITIVEN,
   w        IN OUT  UTL_NLA_ARRAY_DBL,
   z        IN OUT  UTL_NLA_ARRAY_DBL,
   ldz       IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_SBEVD (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   kd       IN      NATURALN,
   ab       IN OUT  UTL_NLA_ARRAY_FLT,
   ldab     IN      POSITIVEN,
   w        IN OUT  UTL_NLA_ARRAY_FLT,
   z        IN OUT  UTL_NLA_ARRAY_FLT,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

Parameters

Table 232-52 LAPACK_SBEVD Procedure Parameters

Parameter Description

jobz

  • 'N': Compute eigenvalues only.

  • 'V': Compute eigenvalues and eigenvectors.

uplo

  • 'U': Upper triangle of A is stored.

  • 'L': Lower triangle of A is stored.

n

The order of the matrix a. N >= 0.

kd

The number of superdiagonals of the matrix A if uplo = 'U', or the number of subdiagonals if uplo = 'L'. kd >= 0.

ab

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldab, n).

On entry, the upper or lower triangle of the symmetric band matrix A stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array ab:

  • If uplo = 'U', ab(kd+1+i-j,j) = a(i,j) for max(1,j-kd)<=i<=j.

  • If uplo = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

On exit, ab is overwritten by values generated during the reduction to tridiagonal form:

  • If uplo = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T are returned in rows kd and kd+1 of ab.

  • If uplo = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of ab.

ldab

The leading dimension of the array ab. ldab >= kd + 1.

w

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldz,n).

If info = 0, the eigenvalues in ascending order.

z

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

  • If jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors of the matrix A, with the i-th column of z holding the eigenvector associated with w(i).

  • If jobz = 'N', then z is not referenced.

ldz

The leading dimension of the array z. ldz >= 1, and if jobz = 'v', ldz >= max(1,n).

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_SPEV Procedures

This procedure computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_SPEV (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   ap       IN OUT  UTL_NLA_ARRAY_DBL,
   w        IN OUT  UTL_NLA_ARRAY_DBL,
   z        IN OUT  UTL_NLA_ARRAY_DBL,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_SPEV (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   ap       IN OUT  UTL_NLA_ARRAY_FLT,
   w        IN OUT  UTL_NLA_ARRAY_FLT,
   z        IN OUT  UTL_NLA_ARRAY_FLT,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

Parameters

Table 232-53 LAPACK_SPEV Procedure Parameters

Parameter Description

jobz

  • 'N': Compute eigenvalues only.

  • 'V': Compute eigenvalues and eigenvectors.

uplo

  • 'U': Upper triangle of A is stored.

  • 'L': Lower triangle of A is stored.

n

The order of the matrix a. N >= 0.

ap

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n*(n+1)/2).

On entry, the upper or lower triangle of the symmetric matrix a packed columnwise in a linear array. The j-th column of a is stored in the array ap:

  • If uplo = 'U', ap(i + (j-1)*j/2) = a(i,j) for 1<=i<=j.

  • If uplo = 'L', ap(i + (j-1)*(2*n-j)/2) = a(i,j) for j<=i<=n.

On exit, ap is overwritten by values generated during the reduction to tridiagonal form:

  • If uplo = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A.

  • If uplo = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.

w

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

If info = 0, the eigenvalues in ascending order.

z

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldz,n).

  • If jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors of the matrix A, with the i-th column of z holding the eigenvector associated with w(i).

  • If jobz = 'N', then z is not referenced.

ldz

The leading dimension of the array z. ldz >= 1, and if jobz = 'v', ldz >= max(1,n).

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_SPEVD Procedures

This procedure computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm makes very mild assumptions about floating point arithmetic.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_SPEVD (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   ap       IN OUT  UTL_NLA_ARRAY_DBL,
   w        IN OUT  UTL_NLA_ARRAY_DBL,
   z        IN OUT  UTL_NLA_ARRAY_DBL,
   ldz       IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_SPEVD (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   ap       IN OUT  UTL_NLA_ARRAY_FLT,
   w        IN OUT  UTL_NLA_ARRAY_FLT,
   z        IN OUT  UTL_NLA_ARRAY_FLT,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

Parameters

Table 232-54 LAPACK_SPEVD Procedure Parameters

Parameter Description

jobz

  • 'N': Compute eigenvalues only.

  • 'V': Compute eigenvalues and eigenvectors.

uplo

  • 'U': Upper triangle of A is stored.

  • 'L': Lower triangle of A is stored.

n

The order of the matrix a. N >= 0.

ap

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n*(n+1)/2).

On entry, the upper or lower triangle of the symmetric matrix a packed columnwise in a linear array. The j-th column of a is stored in the array ap:

  • If uplo = 'U', ap(i + (j-1)*j/2) = a(i,j) for 1<=i<=j.

  • If uplo = 'L', ap(i + (j-1)*(2*n-j)/2) = a(i,j) for j<=i<=n.

On exit, ap is overwritten by values generated during the reduction to tridiagonal form:

  • If uplo = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A.

  • If uplo = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.

w

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

If info = 0, the eigenvalues in ascending order.

z

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldz,n).

  • If jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors of the matrix A, with the i-th column of z holding the eigenvector associated with w(i).

  • If jobz = 'N', then z is not referenced.

ldz

The leading dimension of the array z. ldz >= 1, and if jobz = 'v', ldz >= max(1,n).

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_SPSV Procedures

This procedure computes the solution to a real system of linear equations

a * x = b

where a is an n by n symmetric matrix stored in packed format, and x and b are n by nrhs matrices.

The diagonal pivoting method is used to factor A as

A = U * D * U**T, if UPLO = 'U'

or

A = L * D * L**T, if UPLO = 'L'

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1 by 1 and 2 by 2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B.

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_SPSV (
   uplo    IN      flag,
   n       IN      POSITIVEN,
   nrhs    IN      POSITIVEN,
   ap      IN OUT  UTL_NLA_ARRAY_DBL,
   ipiv    IN OUT  UTL_NLA_ARRAY_INT,
   b       IN OUT  UTL_NLA_ARRAY_DBL,
   ldb     IN      POSITIVEN,
   info    OUT     INTEGER,
   pack    IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_SPSV (
   uplo   IN       flag,
   n      IN       POSITIVEN,
   nrhs   IN       POSITIVEN,
   ap     IN OUT   UTL_NLA_ARRAY_FLT,
   ipiv   IN OUT   UTL_NLA_ARRAY_INT,
   b      IN OUT   UTL_NLA_ARRAY_FLT,
   ldb    IN       POSITIVEN,
   info   OUT      INTEGER,
   pack   IN       flag DEFAULT 'C');

Parameters

Table 232-55 LAPACK_SPSV Procedure Parameters

Parameter Description

uplo

  • uplo = 'U'. Upper triangular of A is stored.

  • uplo = 'L' . Lower triangular of A is stored.

n

The number of linear equations, which is the order of the matrix a. N >= 0.

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

ap

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n*(n+1)/2).

On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array ap as follows:

  • uplo = 'U': AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j

  • uplo = 'L' : AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n

See below for further details.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as a packed triangular matrix in the same storage format as A.

ipiv

INTEGER array, DIMENSION (n).

Details of the interchanges and the block structure of d, as determined by SSPTRF.

  • If ipiv(k) > 0, then rows and columns k and ipiv(k) were interchanged, and d(k,k) is a 1 by1 diagonal block.

  • If uplo = 'U' and ipiv(k) = ipiv(k-1) < 0, then rows and columns k-1 and -ipiv(k) were interchanged and d(k-1:k,k-1:k) is a 2 by 2 diagonal block.

  • If uplo = 'L' and ipiv(k) = ipiv(k+1) < 0, then rows and columns k+1 and -ipiv(k) were interchanged and d(k:k+1,k:k+1) is a 2 by 2 diagonal block.

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the n by nrhs right hand side matrix b.

On exit, if info = 0 , the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max(1,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, d(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix d is exactly singular, so the solution could not be computed.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_STEV Procedures

This procedure computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_STEV (
   jobz     IN      flag,
   n        IN      POSITIVEN,
   d        IN OUT  UTL_NLA_ARRAY_DBL,
   e        IN OUT  UTL_NLA_ARRAY_DBL,
   z        IN OUT  UTL_NLA_ARRAY_DBL,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_STEV (
   jobz     IN      flag,
   n        IN      POSITIVEN,
   d        IN OUT  UTL_NLA_ARRAY_FLT,
   e        IN OUT  UTL_NLA_ARRAY_FLT,
   z        IN OUT  UTL_NLA_ARRAY_FLT,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

Parameters

Table 232-56 LAPACK_STEV Procedure Parameters

Parameter Description

jobz

  • 'N': Compute eigenvalues only.

  • 'V': Compute eigenvalues and eigenvectors.

n

The order of the matrix a. N >= 0.

d

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

  • On entry, the n diagonal elements of the tridiagonal matrix A.

  • On exit, if info = 0, the eigenvalues in ascending order.

e

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

  • On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A, stored in elements 1 to n-1 of e. e(n) need not be set, but is used by the subprogram.

  • On exit, the contents of e are destroyed.

z

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldz, n).

  • If jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors of the matrix A, with the i-th column of z holding the eigenvector associated with d(i).

  • If jobz = 'N', then z is not referenced.

ldz

The leading dimension of the array z. ldz >= 1, and if jobz = 'v', ldz >= max(1,n).

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_STEVD Procedures

This procedure computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_STEVD (
   jobz     IN      flag,
   n        IN      POSITIVEN,
   d        IN OUT  UTL_NLA_ARRAY_DBL,
   e        IN OUT  UTL_NLA_ARRAY_DBL,
   z        IN OUT  UTL_NLA_ARRAY_DBL,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_STEVD(
   jobz     IN      flag,
   n        IN      POSITIVEN,
   d        IN OUT  UTL_NLA_ARRAY_FLT,
   e        IN OUT  UTL_NLA_ARRAY_FLT,
   z        IN OUT  UTL_NLA_ARRAY_FLT,
   ldz      IN      POSITIVEN,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

Parameters

Table 232-57 LAPACK_STEVD Procedure Parameters

Parameter Description

jobz

  • 'N': Compute eigenvalues only.

  • 'V': Compute eigenvalues and eigenvectors.

n

The order of the matrix a. N >= 0.

d

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

  • On entry, the n diagonal elements of the tridiagonal matrix A.

  • On exit, if info = 0, the eigenvalues in ascending order.

e

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

  • On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A, stored in elements 1 to n-1 of e. e(n) need not be set, but is used by the subprogram.

  • On exit, the contents of e are destroyed.

z

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldz, n).

  • If jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors of the matrix A, with the i-th column of z holding the eigenvector associated with d(i).

  • If jobz = 'N', then z is not referenced.

ldz

The leading dimension of the array z. ldz >= 1, and if jobz = 'v', ldz >= max(1,n).

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_SYEV Procedures

This procedure computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_SYEV (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   a        IN OUT  UTL_NLA_ARRAY_DBL,
   lda      IN      POSITIVEN,
   w        IN OUT  UTL_NLA_ARRAY_DBL,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_SYEV (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   a        IN OUT  UTL_NLA_ARRAY_FLT,
   lda      IN      POSITIVEN,
   w        IN OUT  UTL_NLA_ARRAY_FLT,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

Parameters

Table 232-58 LAPACK_SYEV Procedure Parameters

Paramete Description

jobz

  • 'N': Compute eigenvalues only.

  • 'V': Compute eigenvalues and eigenvectors.

uplo

  • 'U': Upper triangle of A is stored.

  • 'L': Upper triangle of A is stored.

n

The order of the matrix a. N >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

On entry, the symmetric matrix a:

  • If uplo = 'U', the leading n by n upper triangular part of a contains the upper triangular part of the matrix a.

  • If uplo = 'L', the leading n byn lower triangular part of a contains the lower triangular part of the matrix a.

On exit:

  • If jobz = 'V', then if info = 0, a contains the orthonormal eigenvectors of the matrix a.

  • If jobz = 'N', then on exit the lower triangle (if uplo = 'L') or the upper triangle (if uplo='U') of a, including the diagonal, is destroyed.

lda

The leading dimension of the array a. lda >= max(1,n).

w

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

If info = 0, the eigenvalues in ascending order.

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_SYEVD Procedures

This procedure computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm that makes mild assumptions about floating point arithmetic.

See Also:

LAPACK Driver Routines (LLS and Eigenvalue Problems) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_SYEVD (

   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   a        IN OUT  UTL_NLA_ARRAY_DBL,
   lda      IN      POSITIVEN,
   w        IN OUT  UTL_NLA_ARRAY_DBL,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_SYEVD (
   jobz     IN      flag,
   uplo     IN      flag,
   n        IN      POSITIVEN,
   a        IN OUT  UTL_NLA_ARRAY_FLT,
   lda      IN      POSITIVEN,
   w        IN OUT  UTL_NLA_ARRAY_FLT,
   info     OUT     INTEGER,
   pack     IN      flag DEFAULT 'C');

Parameters

Table 232-59 LAPACK_SYEVD Procedure Parameters

Parameter Description

jobz

  • 'N': Compute eigenvalues only.

  • 'V': Compute eigenvalues and eigenvectors.

uplo

  • 'U' : Upper triangle of A is stored.

  • 'L': Upper triangle of A is stored.

n

The order of the matrix a. N >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (lda, n).

On entry, the symmetric matrix a:

  • If uplo = 'U', the leading n by n upper triangular part of a contains the upper triangular part of the matrix a.

  • If uplo = 'L', the leading n byn lower triangular part of a contains the lower triangular part of the matrix a.

On exit:

  • If jobz = 'V', then if info = 0, a contains the orthonormal eigenvectors of the matrix a.

  • If jobz = 'N', then on exit the lower triangle (if uplo = 'L') or the upper triangle (if uplo = 'U') of a, including the diagonal, is destroyed.

lda

The leading dimension of the array a. lda >= max(1,n).

w

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n).

If info = 0, the eigenvalues in ascending order.

info

  • = 0 : successful exit

  • < 0 : if info = -i, the i-th argument had an illegal value

  • > 0 : if info = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero

pack

(Optional) Flags the packing of the matricies:

  • 'C': column-major (default)

  • 'R': row-major

LAPACK_SYSV Procedures

This procedure computes the solution to a real system of linear equations

a * x = b

where a is an n by n symmetric matrix, and x and b are n by nrhs matrices.

The diagonal pivoting method is used to factor A as

A = U * D * U**T, if UPLO = 'U'

or

A = L * D * L**T, if UPLO = 'L'

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1 by 1 and 2 by 2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B.

See Also:

LAPACK Driver Routines (Linear Equations) Subprograms for other subprograms in this group

Syntax

UTL_NLA.LAPACK_SYSV (
   uplo    IN      flag,
   n       IN      POSITIVEN,
   nrhs    IN      POSITIVEN,
   a       IN OUT  UTL_NLA_ARRAY_DBL,
   lda     IN      POSITIVEN,
   ipiv    IN OUT  UTL_NLA_ARRAY_INT,
   b       IN OUT  UTL_NLA_ARRAY_DBL,
   ldb     IN      POSITIVEN,
   info    OUT     INTEGER,
   pack    IN      flag DEFAULT 'C');

UTL_NLA.LAPACK_SYSV (
   uplo   IN       flag,
   n      IN       POSITIVEN,
   nrhs   IN       POSITIVEN,
   a      IN OUT   UTL_NLA_ARRAY_FLT,
   lda    IN       POSITIVEN,
   ipiv   IN OUT   UTL_NLA_ARRAY_INT,
   b      IN OUT   UTL_NLA_ARRAY_FLT,
   ldb    IN       POSITIVEN,
   info   OUT      INTEGER,
   pack   IN       flag DEFAULT 'C');

Parameters

Table 232-60 LAPACK_SYSV Procedure Parameters

Parameter Description

uplo

  • uplo = 'U'. Upper triangular of A is stored.

  • uplo = 'L' . Lower triangular of A is stored.

n

The number of linear equations, which is the order of the matrix a. N >= 0.

nrhs

The number of right-hand sides, which is the number of columns of the matrix b. nrhs >= 0.

a

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (n-1).

On entry, the symmetric matrix a. If UPLO = 'U', the leading n by n upper triangular part of a contains the upper triangular part of the matrix a, and the strictly lower triangular part of a is not referenced. If uplo = 'L', the leading n by n lower triangular part of a contains the lower triangular part of the matrix a, and the strictly upper triangular part of a is not referenced.

On exit, if info = 0, the block diagonal matrix d and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by SSYTRF.

lda

The leading dimension of the array a.

lda >= max(1,n)

ipiv

INTEGER array, DIMENSION (ldb, nrhs).

Details of the interchanges and the block structure of d, as determined by SSYTRF.

  • If ipiv(k) > 0, then rows and columns k and ipiv(k) were interchanged, and d(k,k) is a 1 by1 diagonal block.

  • If uplo = 'U' and ipiv(k) = ipiv(k-1) < 0, then rows and columns k-1 and -ipiv(k) were interchanged and d(k-1:k,k-1:k) is a 2 by 2 diagonal block.

  • If uplo = 'L' and ipiv(k) = ipiv(k+1) < 0, then rows and columns k+1 and -ipiv(k) were interchanged and d(k:k+1,k:k+1) is a 2 by 2 diagonal block.

b

UTL_NLA_ARRAY_FLT/DBL, DIMENSION (ldb, nrhs).

On entry, the n by nrhs matrix of right hand side matrix b.

On exit, if info = 0 , the n by nrhs solution matrix X.

ldb

The leading dimension of the array b.

ldb >= max(1,n)

info

  • = 0 : successful exit

  • < 0 : if info = -i , the i-th argument had an illegal value

  • > 0 : if info = i, d(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix d is exactly singular, so the solution could not be computed.

pack

(Optional) Flags the packing of the matrices:

  • 'C': column-major (default)

  • 'R': row-major