## table of contents

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Contributors: **

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Contributors: **

## 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 = ( 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

**Contributors: **

**References: **

[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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