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complexGEauxiliary(3) LAPACK complexGEauxiliary(3)

NAME

complexGEauxiliary

SYNOPSIS

Functions


subroutine cgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE)
CGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2. subroutine cgetc2 (N, A, LDA, IPIV, JPIV, INFO)
CGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix. real function clange (NORM, M, N, A, LDA, WORK)
CLANGE 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 claqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
CLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ. subroutine ctgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, INFO)
CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

Detailed Description

This is the group of complex auxiliary functions for GE matrices

Function Documentation

subroutine cgesc2 (integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) RHS, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, real SCALE)

CGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.

Purpose:

 CGESC2 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 CGETC2.

Parameters:

N

          N is INTEGER
          The number of columns of the matrix A.

A

          A is COMPLEX array, dimension (LDA, N)
          On entry, the  LU part of the factorization of the n-by-n
          matrix A computed by CGETC2:  A = P * L * U * Q

LDA

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

RHS

          RHS is COMPLEX array, dimension N.
          On entry, the right hand side vector b.
          On exit, the solution vector X.

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).

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).

SCALE

          SCALE 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:

December 2016

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

subroutine cgetc2 (integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer INFO)

CGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.

Purpose:

 CGETC2 computes an LU factorization, using 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 a level 1 BLAS version of the algorithm.

Parameters:

N

          N is INTEGER
          The order of the matrix A. N >= 0.

A

          A is COMPLEX array, dimension (LDA, N)
          On entry, the n-by-n matrix 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, giving a nonsingular perturbed system.

LDA

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

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).

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).

INFO

          INFO is INTEGER
           = 0: successful exit
           > 0: if INFO = k, U(k, k) is likely to produce overflow if
                one tries 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, S-901 87 Umea, Sweden.

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

CLANGE 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:

 CLANGE  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 complex matrix A.

Returns:

CLANGE

    CLANGE = ( 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 CLANGE as described
          above.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.  When M = 0,
          CLANGE is set to zero.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0.  When N = 0,
          CLANGE is set to zero.

A

          A is COMPLEX array, dimension (LDA,N)
          The m by n matrix A.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

WORK

          WORK 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:

December 2016

subroutine claqge (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) R, real, dimension( * ) C, real ROWCND, real COLCND, real AMAX, character EQUED)

CLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.

Purpose:

 CLAQGE 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.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0.

A

          A is COMPLEX 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.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

R

          R is REAL array, dimension (M)
          The row scale factors for A.

C

          C is REAL array, dimension (N)
          The column scale factors for A.

ROWCND

          ROWCND is REAL
          Ratio of the smallest R(i) to the largest R(i).

COLCND

          COLCND is REAL
          Ratio of the smallest C(i) to the largest C(i).

AMAX

          AMAX is REAL
          Absolute value of largest matrix entry.

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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

subroutine ctgex2 (logical WANTQ, logical WANTZ, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( ldz, * ) Z, integer LDZ, integer J1, integer INFO)

CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

Purpose:

 CTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22)
 in an upper triangular matrix pair (A, B) by an unitary equivalence
 transformation.
 (A, B) must be in generalized Schur canonical form, that is, A and
 B are both upper triangular.
 Optionally, the matrices Q and Z of generalized Schur vectors are
 updated.
        Q(in) * A(in) * Z(in)**H = Q(out) * A(out) * Z(out)**H
        Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H

Parameters:

WANTQ

          WANTQ is LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

WANTZ

          WANTZ is LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

N

          N is INTEGER
          The order of the matrices A and B. N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the matrix A in the pair (A, B).
          On exit, the updated matrix A.

LDA

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

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the matrix B in the pair (A, B).
          On exit, the updated matrix B.

LDB

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

Q

          Q is COMPLEX array, dimension (LDQ,N)
          If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
          the updated matrix Q.
          Not referenced if WANTQ = .FALSE..

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= 1;
          If WANTQ = .TRUE., LDQ >= N.

Z

          Z is COMPLEX array, dimension (LDZ,N)
          If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit,
          the updated matrix Z.
          Not referenced if WANTZ = .FALSE..

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= 1;
          If WANTZ = .TRUE., LDZ >= N.

J1

          J1 is INTEGER
          The index to the first block (A11, B11).

INFO

          INFO is INTEGER
           =0:  Successful exit.
           =1:  The transformed matrix pair (A, B) would be too far
                from generalized Schur form; the problem is ill-
                conditioned.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

June 2017

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, S-901 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 Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. [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, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996.

Author

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