# sglaux - Man Page

## Name

sglaux — Single Precision Auxiliary Routines

— Helper routines for single precision computations.

## Synopsis

### Functions

subroutine **sla_itranspose** (k, kh, l, lh, x, ldx)

Solver for 4-by-4 Linear Systems.

subroutine **sla_qtrmm2** (side, trans, m, n, alpha, a, lda, b, ldb, c, ldc)

Multiply with the sub-diagonal entries of a quasi triangular matrix.

subroutine **sla_small_solve4** (n, a, rhs, info, eps, smlnum)

Solver for 4-by-4 Linear Systems.

subroutine **sla_small_solve8** (n, a, rhs, info, eps, smlnum)

Solver for 8-by-8 Linear Systems.

subroutine **sla_sort_ev** (n, a, lda, q, ldq, nb, work, info)

Block Reordering of generalized Eigenvalues.

subroutine **sla_sort_gev** (n, a, lda, b, ldb, q, ldq, z, ldz, nb, work, lwork, info)

Block Reordering of generalized Eigenvalues.

subroutine **sla_transform_general** (trans, m, n, x, ldx, qa, ldqa, qb, ldqb, work)

Transform the Right-Hand Side or Solution for generalized projections.

subroutine **sla_transform_standard** (trans, m, x, ldx, q, ldq, work)

Transform the Right-Hand Side or Solution for standard projections.

## Detailed Description

Helper routines for single precision computations.

This section contains a set of helper routines for the single precision computational routines. This includes the transformation of the right-hand-sides, the transformation of the solutions, the solution of special linear systems and the sorting of eigenvalues.

## Function Documentation

### subroutine sla_itranspose (integer k, integer kh, integer l, integer lh, real, dimension(ldx,*) x, integer ldx)

Solver for 4-by-4 Linear Systems.

**Purpose:**

SLA_ITRANSPOSE performs the implicit transpose X(L:LH,K:KH ) <- X(K:KH,L:LH)**T (1)

**Parameters***K*K is INTEGER First index of the transpose operations as in (1)

*KH*KH is INTEGER Second index of the transpose operations as in (1)

*L*L is INTEGER Third index of the transpose operations as in (1)

*LH*LH is INTEGER Fourth index of the transpose operations as in (1)

*X*X is REAL array, dimension (LDX,*) X is the matrix to perform the transpose on.

*LDX*LDX is INTEGER The leading dimension of the array X. LDX >= MAX(1,KH,LH).

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **86** of file **sla_itranspose.f90**.

### subroutine sla_qtrmm2 (character(1) side, character(1) trans, integer m, integer n, real alpha, real, dimension(lda, *) a, integer lda, real, dimension(ldb, *) b, integer ldb, real, dimension(ldc,*) c, integer ldc)

Multiply with the sub-diagonal entries of a quasi triangular matrix.

**Purpose:**

Multiply a matrix with the sub-diagonal entries of a quasi-triangular matrix. Consider a quasi-triangular matrix A where sub(A) is the matrix only containing the sub-diagonal entries. Than the SLA_QTRMM2 routine computes C = C + ALPHA * op(sub(A)) * B (1) or C = C + ALPHA * B * op(sub(A)) (2)

**Attention**The routine does not check the input arguments.

**Parameters***SIDE*SIDE is CHARACTER(1) On entry, SIDE specifies whether op( A ) multiplies B from the left or right as follows: SIDE = 'L' or 'l' B := alpha*op( A )*B. SIDE = 'R' or 'r' B := alpha*B*op( A ).

*TRANS*TRANS is CHARACTER(1) On entry, TRANS specifies the form of op( A ) to be used in the matrix multiplication as follows: TRANS == 'N' or 'n' op( A ) = A. TRANS == 'T' or 't' op( A ) = A**T. TRANS = 'C' or 'c' op( A ) = A**T.

*M*M is INTEGER On entry, M specifies the number of rows of B. M must be at least zero.

*N*N is INTEGER On entry, N specifies the number of columns of B. N must be at least zero.

*ALPHA*ALPHA is REAL. On entry, ALPHA specifies the scalar alpha. When alpha is zero then A is not referenced and B need not be set before entry.

*A*A is REAL array of DIMENSION ( LDA, k ), where k is m when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'. The leading k by k upper triangular part of the array A must contain the upper quasi triangular matrix.

*LDA*LDA is INTEGER On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. When SIDE = 'L' or 'l' then LDA must be at least max( 1, m ), when SIDE = 'R' or 'r' then LDA must be at least max( 1, n ).

*B*B is REAL array of DIMENSION ( LDB, n ). Before entry, the leading m by n part of the array B must contain the matrix B.

*LDB*LDB is INTEGER On entry, LDB specifies the first dimension of B as declared in the calling (sub) program. LDB must be at least max( 1, m ).

*C*C is REAL array of DIMENSION ( LDC, n ). Before entry, the leading m by n part of the array C must contain the matrix C, and on exit is overwritten by the update January 2021

*LDC*LDC is INTEGER On entry, LDC specifies the first dimension of C as declared in the calling (sub) program. LDB must be at least max( 1, m ).

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **153** of file **sla_qtrmm2.f90**.

### subroutine sla_small_solve4 (integer n, real, dimension( 4, 4 ) a, real, dimension(4) rhs, integer info, real eps, real smlnum)

Solver for 4-by-4 Linear Systems.

**Purpose:**

SLA_SMALL_SOLVE4 solves a linear system A * x = b (1) where A is a most a 4 by 4 matrix.

**Parameters***N*N is INTEGER The order of the matrix A N = (1,2,4).

*A*A is REAL array, dimension (4,4) The matrix A must be stored in a 4-by-4 matrix even if it is smaller. On output the matrix A is destroyed.

*RHS*RHS is REAL array, dimension (4) The matrix RHS contains the right hand side on input and the solution X on output if INFO == 0

*INFO*INFO is INTEGER == 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.

*EPS*EPS is REAL EPS is the machine epsilon determined by DLAMCH

*SMLNUM*SMLNUM is REAL SMLNUM is the smallest number in REAL which can be represented without underflow.

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **98** of file **sla_small_solve4.f90**.

### subroutine sla_small_solve8 (integer n, real, dimension( 8, 8 ) a, real, dimension(8) rhs, integer info, real eps, real smlnum)

Solver for 8-by-8 Linear Systems.

**Purpose:**

SLA_SMALL_SOLVE8 solves a linear system A * x = b (1) where A is a most a 8 by 8 matrix with leading dimension 8.

**Parameters***N*N is INTEGER The order of the matrix A N = {1,2,4,8} .

*A*A is REAL array, dimension (8,8) The matrix A must be stored in a 4-by-4 matrix even if it is smaller. On output the matrix A is destroyed.

*RHS*RHS is REAL array, dimension (8) The matrix RHS contains the right hand side on input and the solution X on output if INFO == 0

*INFO*INFO is INTEGER == 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: The equation is not solved correctly. One of the arising inner system got singular.

*EPS*EPS is REAL EPS is the machine epsilon determined by DLAMCH

*SMLNUM*SMLNUM is REAL SMLNUM is the smallest number in REAL which can be represented without underflow.

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **98** of file **sla_small_solve8.f90**.

### subroutine sla_sort_ev (integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldq, * ) q, integer ldq, integer nb, real, dimension( * ) work, integer info)

Block Reordering of generalized Eigenvalues.

**Purpose:**

Reorder the eigenvalues of a matrix A such that the complex eigenvalues pairs will not cause a change of the block size in blocked linear matrix equation solver codes. The function uses STREXC from LAPACK for moving the eigenvalues on the diagonal.

**Parameters***N*N is INTEGER The order of the matrices A and Q, N >= 0.

*A*A is a REAL array, dimension (LDA,N) On entry, the leading N-by-N upper Hessenberg part of this array must contain the generalized Schur factor A_s of the matrix A. On exit, the leading N-by-N part of this array contains the generalized Schur factor A_s of the matrix A with no complex eigenvalues pairs an A(k*NB,k*NB), where k is a non negative integer.

*LDA*LDA is INTEGER The leading dimension of the array A. LDA >= MAX(1,N).

*Q*Q is a REAL array, dimension (LDQ,N) On entry, the leading N-by-N part of this array must contain the orthogonal matrix Q from the generalized Schur factorization. On exit, the leading N-by-N part of this array contains the update January 2021 factorization with integrated eigenvalue reordering.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= MAX(1,N).

*NB*NB is INTEGER Block size of the solver planed to use. Typical values are 16, 32, or 64. The block size must be an even positive integer otherwise an error is returned.

*WORK*WORK is INTEGER array, dimension (N) Workspace

*INFO*INFO is INTEGER == 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: The internal STGEX2 caused an error. ]

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **124** of file **sla_sort_ev.f90**.

### subroutine sla_sort_gev (integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, integer nb, real, dimension( * ) work, integer lwork, integer info)

Block Reordering of generalized Eigenvalues.

**Purpose:**

Reorder the eigenvalues of a matrix pencil (A,B) such that the complex eigenvalues pairs will not cause a change of the block size in blocked linear matrix equation solver codes. The function uses STGEX2 from LAPACK for moving the eigenvalues on the diagonal.

**Parameters***N*N is INTEGER The order of the matrices A, B, Q, and Z, N >= 0.

*A*A is a REAL array, dimension (LDA,N) On entry, the leading N-by-N upper Hessenberg part of this array must contain the generalized Schur factor A_s of the matrix A. On exit, the leading N-by-N part of this array contains the generalized Schur factor A_s of the matrix A with no complex eigenvalues pairs an A(k*NB,k*NB), where k is a non negative integer.

*LDA*LDA is INTEGER The leading dimension of the array A. LDA >= MAX(1,N).

*B*B is a REAL array, dimension (LDB,N) On entry, the leading N-by-N upper triangular part of this array must contain the generalized Schur factor B_s of the matrix B, On exit, the leading N-by-N part of this array contains the generalized Schur factor B_s of the matrix B with reordered eigenvalues.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= MAX(1,N).

*Q*Q is a REAL array, dimension (LDQ,N) On entry, the leading N-by-N part of this array must contain the orthogonal matrix Q from the generalized Schur factorization. On exit, the leading N-by-N part of this array contains the update January 2021 factorization with integrated eigenvalue reordering.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= MAX(1,N).

*Z*Z is a REAL array, dimension (LDZ,N) On entry, the leading N-by-N part of this array must contain the orthogonal matrix Z from the generalized Schur factorization. On exit, the leading N-by-N part of this array contains the update January 2021 factorization with integrated eigenvalue reordering.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= MAX(1,N).

*NB*NB is INTEGER Block size of the solver planed to use. Typical values are 16, 32, or 64. The block size must be an even positive integer otherwise an error is returned.

*WORK*WORK is INTEGER array, dimension (LWORK) Workspace

*LWORK*LWORK is INTEGER Size of the workspace. LWORK >= MAX(1, 4*N, 256)

*INFO*INFO is INTEGER == 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: The internal STGEX2 caused an error. ]

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **165** of file **sla_sort_gev.f90**.

### subroutine sla_transform_general (character, dimension(*) trans, integer m, integer n, real, dimension(ldx, *) x, integer ldx, real, dimension(ldqa, *) qa, integer ldqa, real, dimension(ldqb, *) qb, integer ldqb, real, dimension(*) work)

Transform the Right-Hand Side or Solution for generalized projections.

**Purpose:**

SLA_TRANSFORM_STANDARD computes either X <- QA*X*QB**T (1) or X <- QA**T*X*QB (2) where QA is a M-by-M matrices, QB is a N-by-N matrix, and X is a M-by-N matrix.

**Remarks**The function does not perform any check on the input arguments.

**Parameters***TRANS*TRANS is CHARACTER(1) Specifies which transformation is applied: == 'N': Transformation (1) == 'T': Transformation (2)

*M*M is INTEGER The order of the matrices QA and the number of rows of X. M >= 0.

*N*N is INTEGER The order of the matrices QB and the number of columns of X. N >= 0.

*X*X is REAL array, dimension (LDX,N) The matrix X is a general M-by-N matrix overwritten by (1) or (2)

*LDX*LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).

*QA*QA is REAL array, dimension (LDQA,M) The matrix QA is M-by-M the transformation matrix.

*LDQA*LDQA is INTEGER The leading dimension of the array QA. LDQA >= max(1,M).

*QB*QB is REAL array, dimension (LDQB,N) The matrix QB is N-by-N the transformation matrix.

*LDQB*LDQB is INTEGER The leading dimension of the array QB. LDQB >= max(1,N).

*WORK*WORK is REAL array, dimension (M*N) Workspace for the computation.

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **119** of file **sla_transform_general.f90**.

### subroutine sla_transform_standard (character, dimension(*) trans, integer m, real, dimension(ldx, *) x, integer ldx, real, dimension(ldq, *) q, integer ldq, real, dimension(*) work)

Transform the Right-Hand Side or Solution for standard projections.

**Purpose:**

SLA_TRANSFORM_STANDARD computes either X <- Q*X*Q**T (1) or X <- Q**T*X*Q (2) where Q and X are M-by-M matrices.

**Remarks**The function does not perform any check on the input arguments.

**Parameters***TRANS*TRANS is CHARACTER(1) Specifies which transformation is applied: == 'N': Transformation (1) == 'T': Transformation (2)

*M*M is INTEGER The order of the matrices Q and X. M >= 0.

*X*X is REAL array, dimension (LDX,M) The matrix X is a general M-by-M matrix overwritten by (1) or (2)

*LDX*LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M).

*Q*Q is REAL array, dimension (LDQ,M) The matrix Q is M-by-M the transformation matrix.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,M).

*WORK*WORK is REAL array, dimension (M**2) Workspace for the computation.

**Author**Martin Koehler, MPI Magdeburg

**Date**Januar 2023

Definition at line **101** of file **sla_transform_standard.f90**.

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