 LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
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## Functions/Subroutines

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 n-by-n matrix.
REAL function slange (NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, 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/Subroutine 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 N-by-N matrix A using the LU factorization with
complete pivoting computed by SGETC2.```
Parameters:
 [in] N ``` N is INTEGER The order of the matrix A.``` [in] A ``` A is REAL array, dimension (LDA,N) On entry, the LU part of the factorization of the n-by-n matrix A computed by SGETC2: A = P * L * U * Q``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1, N).``` [in,out] RHS ``` RHS is REAL array, dimension (N). On entry, the right hand side vector b. On exit, the solution vector X.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).``` [in] JPIV ``` JPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).``` [out] SCALE ``` SCALE is REAL On exit, SCALE contains the scale factor. SCALE is chosen 0 <= SCALE <= 1 to prevent owerflow in the solution.```
Date:
September 2012
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

Definition at line 115 of file sgesc2.f.

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 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 n-by-n matrix.

Purpose:
``` SGETC2 computes an LU factorization with complete pivoting of the
n-by-n 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:
 [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] A ``` A is REAL array, dimension (LDA, N) On entry, the n-by-n 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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] IPIV ``` IPIV is INTEGER array, dimension(N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).``` [out] JPIV ``` JPIV is INTEGER array, dimension(N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).``` [out] INFO ``` INFO 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.```
Date:
September 2012
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

Definition at line 112 of file sgetc2.f.

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 REAL function slange ( character NORM, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK )

SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, 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:
 [in] NORM ``` NORM is CHARACTER*1 Specifies the value to be returned in SLANGE as described above.``` [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0. When M = 0, SLANGE is set to zero.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0. When N = 0, SLANGE is set to zero.``` [in] A ``` A is REAL array, dimension (LDA,N) The m by n matrix A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(M,1).``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced.```
Date:
September 2012

Definition at line 115 of file slange.f.

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 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:
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in,out] A ``` A 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.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(M,1).``` [in] R ``` R is REAL array, dimension (M) The row scale factors for A.``` [in] C ``` C is REAL array, dimension (N) The column scale factors for A.``` [in] ROWCND ``` ROWCND is REAL Ratio of the smallest R(i) to the largest R(i).``` [in] COLCND ``` COLCND is REAL Ratio of the smallest C(i) to the largest C(i).``` [in] AMAX ``` AMAX is REAL Absolute value of largest matrix entry.``` [out] EQUED ``` EQUED 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.```
Date:
September 2012

Definition at line 142 of file slaqge.f.

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 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 1-by-1 or 2-by-2 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 1-by-1 and 2-by-2
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:
 [in] WANTQ ``` WANTQ is LOGICAL .TRUE. : update the left transformation matrix Q; .FALSE.: do not update Q.``` [in] WANTZ ``` WANTZ is LOGICAL .TRUE. : update the right transformation matrix Z; .FALSE.: do not update Z.``` [in] N ``` N is INTEGER The order of the matrices A and B. N >= 0.``` [in,out] A ``` A is REAL arrays, dimensions (LDA,N) On entry, the matrix A in the pair (A, B). On exit, the updated matrix A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] B ``` B is REAL arrays, dimensions (LDB,N) On entry, the matrix B in the pair (A, B). On exit, the updated matrix B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [in,out] Q ``` Q 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..``` [in] LDQ ``` LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N.``` [in,out] Z ``` Z 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..``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N.``` [in] J1 ``` J1 is INTEGER The index to the first block (A11, B11). 1 <= J1 <= N.``` [in] N1 ``` N1 is INTEGER The order of the first block (A11, B11). N1 = 0, 1 or 2.``` [in] N2 ``` N2 is INTEGER The order of the second block (A22, B22). N2 = 0, 1 or 2.``` [out] WORK ` WORK is REAL array, dimension (MAX(1,LWORK)).` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )``` [out] INFO ``` INFO 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 ill-conditioned. <0: If INFO = -16: LWORK is too small. Appropriate value for LWORK is returned in WORK(1).```
Date:
September 2012
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.  for details.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
References:
```   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
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.

 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, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.```

Definition at line 221 of file stgex2.f.

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