Functions
subroutine sgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE)
SGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
subroutine sgetc2 (N, A, LDA, IPIV, JPIV, INFO)
SGETC2 computes the LU factorization with complete pivoting of the general nbyn matrix.
real function slange (NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1norm, Frobenius norm, infinitynorm, or the largest absolute value of any element of a general rectangular matrix.
subroutine slaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
subroutine stgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO)
STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.
Detailed Description
This is the group of real auxiliary functions for GE matrices
Function Documentation
subroutine sgesc2 (integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) RHS, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, real SCALE)
SGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.
Purpose:

SGESC2 solves a system of linear equations A * X = scale* RHS with a general NbyN matrix A using the LU factorization with complete pivoting computed by SGETC2.
Parameters:

N
N is INTEGER The order of the matrix A.
AA is REAL array, dimension (LDA,N) On entry, the LU part of the factorization of the nbyn matrix A computed by SGETC2: A = P * L * U * Q
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1, N).
RHSRHS is REAL array, dimension (N). On entry, the right hand side vector b. On exit, the solution vector X.
IPIVIPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).
JPIVJPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).
SCALESCALE is REAL On exit, SCALE contains the scale factor. SCALE is chosen 0 <= SCALE <= 1 to prevent owerflow in the solution.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
Contributors:
 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901 87 Umea, Sweden.
subroutine sgetc2 (integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer INFO)
SGETC2 computes the LU factorization with complete pivoting of the general nbyn matrix.
Purpose:

SGETC2 computes an LU factorization with complete pivoting of the nbyn matrix A. The factorization has the form A = P * L * U * Q, where P and Q are permutation matrices, L is lower triangular with unit diagonal elements and U is upper triangular. This is the Level 2 BLAS algorithm.
Parameters:

N
N is INTEGER The order of the matrix A. N >= 0.
AA is REAL array, dimension (LDA, N) On entry, the nbyn matrix A to be factored. On exit, the factors L and U from the factorization A = P*L*U*Q; the unit diagonal elements of L are not stored. If U(k, k) appears to be less than SMIN, U(k, k) is given the value of SMIN, i.e., giving a nonsingular perturbed system.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
IPIVIPIV is INTEGER array, dimension(N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).
JPIVJPIV is INTEGER array, dimension(N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).
INFOINFO is INTEGER = 0: successful exit > 0: if INFO = k, U(k, k) is likely to produce owerflow if we try to solve for x in Ax = b. So U is perturbed to avoid the overflow.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 June 2016
Contributors:
 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901 87 Umea, Sweden.
real function slange (character NORM, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)
SLANGE returns the value of the 1norm, Frobenius norm, infinitynorm, or the largest absolute value of any element of a general rectangular matrix.
Purpose:

SLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real matrix A.
Returns:

SLANGE
SLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters:

NORM
NORM is CHARACTER*1 Specifies the value to be returned in SLANGE as described above.
MM is INTEGER The number of rows of the matrix A. M >= 0. When M = 0, SLANGE is set to zero.
NN is INTEGER The number of columns of the matrix A. N >= 0. When N = 0, SLANGE is set to zero.
AA is REAL array, dimension (LDA,N) The m by n matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(M,1).
WORKWORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
subroutine slaqge (integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, character EQUED)
SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Purpose:

SLAQGE equilibrates a general M by N matrix A using the row and column scaling factors in the vectors R and C.
Parameters:

M
M is INTEGER The number of rows of the matrix A. M >= 0.
NN is INTEGER The number of columns of the matrix A. N >= 0.
AA is REAL array, dimension (LDA,N) On entry, the M by N matrix A. On exit, the equilibrated matrix. See EQUED for the form of the equilibrated matrix.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(M,1).
RR is REAL array, dimension (M) The row scale factors for A.
CC is REAL array, dimension (N) The column scale factors for A.
ROWCNDROWCND is REAL Ratio of the smallest R(i) to the largest R(i).
COLCNDCOLCND is REAL Ratio of the smallest C(i) to the largest C(i).
AMAXAMAX is REAL Absolute value of largest matrix entry.
EQUEDEQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration = 'R': Row equilibration, i.e., A has been premultiplied by diag(R). = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C). = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).
Internal Parameters:

THRESH is a threshold value used to decide if row or column scaling should be done based on the ratio of the row or column scaling factors. If ROWCND < THRESH, row scaling is done, and if COLCND < THRESH, column scaling is done. LARGE and SMALL are threshold values used to decide if row scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, row scaling is done.
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 September 2012
subroutine stgex2 (logical WANTQ, logical WANTZ, 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 J1, integer N1, integer N2, real, dimension( * ) WORK, integer LWORK, integer INFO)
STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.
Purpose:

STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1by1 or 2by2 in an upper (quasi) triangular matrix pair (A, B) by an orthogonal equivalence transformation. (A, B) must be in generalized real Schur canonical form (as returned by SGGES), i.e. A is block upper triangular with 1by1 and 2by2 diagonal blocks. B is upper triangular. Optionally, the matrices Q and Z of generalized Schur vectors are updated. Q(in) * A(in) * Z(in)**T = Q(out) * A(out) * Z(out)**T Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
Parameters:

WANTQ
WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q.
WANTZWANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z.
NN is INTEGER The order of the matrices A and B. N >= 0.
AA is REAL arrays, dimensions (LDA,N) On entry, the matrix A in the pair (A, B). On exit, the updated matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
BB is REAL arrays, dimensions (LDB,N) On entry, the matrix B in the pair (A, B). On exit, the updated matrix B.
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).
QQ is REAL array, dimension (LDZ,N) On entry, if WANTQ = .TRUE., the orthogonal matrix Q. On exit, the updated matrix Q. Not referenced if WANTQ = .FALSE..
LDQLDQ is INTEGER The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N.
ZZ is REAL array, dimension (LDZ,N) On entry, if WANTZ =.TRUE., the orthogonal matrix Z. On exit, the updated matrix Z. Not referenced if WANTZ = .FALSE..
LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N.
J1J1 is INTEGER The index to the first block (A11, B11). 1 <= J1 <= N.
N1N1 is INTEGER The order of the first block (A11, B11). N1 = 0, 1 or 2.
N2N2 is INTEGER The order of the second block (A22, B22). N2 = 0, 1 or 2.
WORKWORK is REAL array, dimension (MAX(1,LWORK)).
LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )
INFOINFO is INTEGER =0: Successful exit >0: If INFO = 1, the transformed matrix (A, B) would be too far from generalized Schur form; the blocks are not swapped and (A, B) and (Q, Z) are unchanged. The problem of swapping is too illconditioned. <0: If INFO = 16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1).
Author:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
 November 2015
Further Details:
 In the current code both weak and strong stability tests are performed. The user can omit the strong stability test by changing the internal logical parameter WANDS to .FALSE.. See ref. [2] for details.
Contributors:
 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901 87 Umea, Sweden.
References:

[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and RealTime Applications, Kluwer Academic Publ. 1993, pp 195218. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF  94.04, Department of Computing Science, Umea University, S901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.
Author
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