?tgex2

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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?tgex2

Swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal/unitary equivalence transformation.

Syntax

call stgex2( wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, n1, n2, work, lwork, info )

call dtgex2( wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, n1, n2, work, lwork, info )

call ctgex2( wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info )

call ztgex2( wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info )

Include Files

mkl.fi

Description

The real routines stgex2/dtgex2 swap 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/dgges), that is, A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.

The complex routines ctgex2/ztgex2 swap 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.

All routines optionally update the matrices Q and Z of generalized Schur vectors:

For real flavors,

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.

For complex flavors,

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.

Input Parameters

wantq

LOGICAL.

If wantq = .TRUE. : update the left transformation matrix Q;

If wantq = .FALSE. : do not update Q.

wantz

LOGICAL.

If wantz = .TRUE. : update the right transformation matrix Z;

If wantz = .FALSE.: do not update Z.

n

INTEGER. The order of the matrices A and B. n≥ 0.

a, b

REAL for stgex2DOUBLE PRECISION for dtgex2

COMPLEX for ctgex2

DOUBLE COMPLEX for ztgex2.

Arrays, DIMENSION (lda, n) and (ldb, n), respectively.

On entry, the matrices A and B in the pair (A, B).

lda

INTEGER. The leading dimension of the array a. lda≥ max(1,n).

ldb

INTEGER. The leading dimension of the array b. ldb≥ max(1,n).

q, z

REAL for stgex2DOUBLE PRECISION for dtgex2

COMPLEX for ctgex2

DOUBLE COMPLEX for ztgex2.

Arrays, DIMENSION (ldq, n) and (ldz, n), respectively.

On entry, if wantq = .TRUE., q contains the orthogonal/unitary matrix Q, and if wantz = .TRUE., z contains the orthogonal/unitary matrix Z.

ldq

INTEGER. The leading dimension of the array q. ldq≥ 1.

If wantq = .TRUE., ldq≥n.

ldz

INTEGER. The leading dimension of the array z. ldz≥ 1.

If wantz = .TRUE., ldz≥n.

j1

INTEGER.

The index to the first block (A11, B11). 1 ≤ j1 ≤ n.

n1

INTEGER. Used with real flavors only. The order of the first block (A11, B11). n1 = 0, 1 or 2.

n2

INTEGER. Used with real flavors only. The order of the second block (A22, B22). n2 = 0, 1 or 2.

work

REAL for stgex2

DOUBLE PRECISION for dtgex2.

Workspace array, DIMENSION(max(1,lwork)). Used with real flavors only.

lwork

INTEGER. The dimension of the array work.

lwork≥max(n*(n2+n1), 2*(n2+n1)²)

Output Parameters

a

On exit, the updated matrix A.

B

On exit, the updated matrix B.

Q

On exit, the updated matrix Q.

Not referenced if wantq = .FALSE..

z

On exit, the updated matrix Z.

Not referenced if wantz = .FALSE..

info

INTEGER.

=0: Successful exit For stgex2/dtgex2: 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. If info = -16: lwork is too small. Appropriate value for lwork is returned in work(1).

For ctgex2/ztgex2:

If info = 1, the transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is ill-conditioned.

Parent topic: LAPACK Auxiliary Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?tgex2