SRC/sgges3.f(3)

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version 3.12.1 · 7 versions
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Hey bestie 💖 welcome to the Library Functions Manual page for SRC/sgges3.f(3). This is LAPACK's blocked-algorithm baddie sgges3, and she's here to compute generalized eigenvalues, Schur form, and optionally Schur vectors. Slay.

✨ NAME ✨

SRC/sgges3.f

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine sgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb,
    sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork,
    bwork, info)

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

💅 Function/Subroutine Documentation 💅

subroutine sgges3 (character jobvsl, character jobvsr, character sort,
    external selctg, integer n, real, dimension( lda, * ) a, integer lda,
    real, dimension( ldb, * ) b, integer ldb, integer sdim, real, dimension(
    * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta,
    real, dimension( ldvsl, * ) vsl, integer ldvsl, real, dimension( ldvsr,
    * ) vsr, integer ldvsr, real, dimension( * ) work, integer lwork,
    logical, dimension( * ) bwork, integer info)

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

🌸 Purpose 🌸

Ok so, for a pair of N-by-N real nonsymmetric matrices (A,B), SGGES3 computes the generalized eigenvalues, the generalized real Schur form (S,T), and optionally the left and/or right matrices of Schur vectors VSL and VSR. That gives you the generalized Schur factorization:

(A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T. The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces).

If you only need the generalized eigenvalues, use the driver SGGEV instead, bestie — it's faster. 🏃‍♀️💨

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It's usually represented as the pair (alpha,beta), because there is a reasonable interpretation for beta=0 or both being zero.

A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be by making the corresponding elements of T have the form:

[  a  0  ]
[  0  b  ]

and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues.

Yeah, that "will be by making" sentence is exactly the energy the source gave us — keep it moving. 💀

🎀 Parameters 🎀

JOBVSL

JOBVSL is CHARACTER*1

  • = 'N': do not compute the left Schur vectors;
  • = 'V': compute the left Schur vectors.

JOBVSR

JOBVSR is CHARACTER*1

  • = 'N': do not compute the right Schur vectors;
  • = 'V': compute the right Schur vectors.

SORT

SORT is CHARACTER*1

Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.

  • = 'N': Eigenvalues are not ordered;
  • = 'S': Eigenvalues are ordered (see SELCTG);

SELCTG

SELCTG is a LOGICAL FUNCTION of three REAL arguments. SELCTG must be declared EXTERNAL in the calling subroutine.

If SORT = 'N', SELCTG is not referenced.

If SORT = 'S', SELCTG is used to select eigenvalues to sort to the top left of the Schur form.

An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected.

Note that in the ill-conditioned case, a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in this case.

N

N is INTEGER

The order of the matrices A, B, VSL, and VSR. N >= 0.

A

A is REAL array, dimension (LDA, N)

On entry, the first of the pair of matrices.

On exit, A has been overwritten by its generalized Schur form S.

LDA

LDA is INTEGER

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

B

B is REAL array, dimension (LDB, N)

On entry, the second of the pair of matrices.

On exit, B has been overwritten by its generalized Schur form T.

LDB

LDB is INTEGER

The leading dimension of B. LDB >= max(1,N).

SDIM

SDIM is INTEGER

If SORT = 'N', SDIM = 0.

If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which SELCTG is true. (Complex conjugate pairs for which SELCTG is true for either eigenvalue count as 2.)

ALPHAR

ALPHAR is REAL array, dimension (N)

ALPHAI

ALPHAI is REAL array, dimension (N)

BETA

BETA is REAL array, dimension (N)

On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j=1,...,N are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio. However, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).

VSL

VSL is REAL array, dimension (LDVSL,N)

If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'.

LDVSL

LDVSL is INTEGER

The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N.

VSR

VSR is REAL array, dimension (LDVSR,N)

If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'.

LDVSR

LDVSR is INTEGER

The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.

WORK

WORK is REAL array, dimension (MAX(1,LWORK))

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER

The dimension of the array WORK.

If N = 0, LWORK >= 1, else LWORK >= 6*N+16.

For good performance, LWORK must generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

BWORK

BWORK is LOGICAL array, dimension (N)

Not referenced if SORT = 'N'.

INFO

INFO is INTEGER

  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • = 1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for j=INFO+1,...,N.
  • > N:
    • =N+1: other than QZ iteration failed in SLAQZ0.
    • =N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy SELCTG=.TRUE. This could also be caused due to scaling.
    • =N+3: reordering failed in STGSEN.

🌟 Author 🌟

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 281 of file sgges3.f.

🌈 Author 🌈

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