doubleGEauxiliary(3) LAPACK doubleGEauxiliary(3)

# NAME¶

doubleGEauxiliary

# SYNOPSIS¶

## Functions¶

subroutine dgesc2 (N, A, LDA, RHS, IPIV, JPIV, SCALE)
DGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2. subroutine dgetc2 (N, A, LDA, IPIV, JPIV, INFO)
DGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix. double precision function dlange (NORM, M, N, A, LDA, WORK)
DLANGE 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 dlaqge (M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
DLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ. subroutine dtgex2 (WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, LDZ, J1, N1, N2, WORK, LWORK, INFO)
DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

# Detailed Description¶

This is the group of double auxiliary functions for GE matrices

# Function Documentation¶

## subroutine dgesc2 (integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) RHS, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, double precision SCALE)¶

DGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed by sgetc2.

Purpose:

```
DGESC2 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 DGETC2.```

Parameters

N

```
N is INTEGER

The order of the matrix A.```

A

```
A is DOUBLE PRECISION array, dimension (LDA,N)

On entry, the  LU part of the factorization of the n-by-n

matrix A computed by DGETC2:  A = P * L * U * Q```

LDA

```
LDA is INTEGER

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

RHS

```
RHS is DOUBLE PRECISION 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 DOUBLE PRECISION

On exit, SCALE contains the scale factor. SCALE is chosen

0 <= SCALE <= 1 to prevent overflow in the solution.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

November 2017

Contributors:

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

## subroutine dgetc2 (integer N, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV, integer INFO)¶

DGETC2 computes the LU factorization with complete pivoting of the general n-by-n matrix.

Purpose:

```
DGETC2 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

N

```
N is INTEGER

The order of the matrix A. N >= 0.```

A

```
A is DOUBLE PRECISION 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.```

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

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

NAG Ltd.

Date

June 2016

Contributors:

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

## double precision function dlange (character NORM, integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) WORK)¶

DLANGE 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:

```
DLANGE  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

DLANGE

```
DLANGE = ( 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 DLANGE as described

above.```

M

```
M is INTEGER

The number of rows of the matrix A.  M >= 0.  When M = 0,

DLANGE is set to zero.```

N

```
N is INTEGER

The number of columns of the matrix A.  N >= 0.  When N = 0,

DLANGE is set to zero.```

A

```
A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK)),

where LWORK >= M when NORM = 'I'; otherwise, WORK is not

referenced.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine dlaqge (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) R, double precision, dimension( * ) C, double precision ROWCND, double precision COLCND, double precision AMAX, character EQUED)¶

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

Purpose:

```
DLAQGE 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (M)

The row scale factors for A.```

C

```
C is DOUBLE PRECISION array, dimension (N)

The column scale factors for A.```

ROWCND

```
ROWCND is DOUBLE PRECISION

Ratio of the smallest R(i) to the largest R(i).```

COLCND

```
COLCND is DOUBLE PRECISION

Ratio of the smallest C(i) to the largest C(i).```

AMAX

```
AMAX is DOUBLE PRECISION

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

NAG Ltd.

Date

December 2016

## subroutine dtgex2 (logical WANTQ, logical WANTZ, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( ldz, * ) Z, integer LDZ, integer J1, integer N1, integer N2, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

Purpose:

```
DTGEX2 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 DGGES), 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

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 DOUBLE PRECISION array, dimensions (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 DOUBLE PRECISION array, dimensions (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 DOUBLE PRECISION array, dimension (LDQ,N)

On entry, if WANTQ = .TRUE., the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ,N)

On entry, if WANTZ =.TRUE., the orthogonal 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). 1 <= J1 <= N.```

N1

```
N1 is INTEGER

The order of the first block (A11, B11). N1 = 0, 1 or 2.```

N2

```
N2 is INTEGER

The order of the second block (A22, B22). N2 = 0, 1 or 2.```

WORK

```
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).```

LWORK

```
LWORK is INTEGER

The dimension of the array WORK.

LWORK >=  MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )```

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).```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

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.```

# Author¶

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