ggesx(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

okay so bestie, welcome to ggesx(3) from the Library Functions Manual — ggesx: Schur form, expert, and she is iconic 💅✨

🎀 NAME 🎀

  • ggesx — ggesx: Schur form, expert.

✨ SYNOPSIS ✨

🌸 Functions 🌸

cggesx

subroutine cggesx (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b,
     ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv,
     work, lwork, rwork, iwork, liwork, bwork, info)

CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices.

dggesx

subroutine dggesx (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b,
     ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde,
     rcondv, work, lwork, iwork, liwork, bwork, info)

DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices.

sggesx

subroutine sggesx (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b,
     ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, rconde,
     rcondv, work, lwork, iwork, liwork, bwork, info)

SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices.

zggesx

subroutine zggesx (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b,
     ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv,
     work, lwork, rwork, iwork, liwork, bwork, info)

ZGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices.

🦋 Detailed Description 🦋

(this section is literally just the heading in this part, bestie — the actual function documentation is right below and it’s giving main character energy)

💖 Function Documentation 💖

🌟 subroutine cggesx 🌟

subroutine cggesx (character jobvsl, character jobvsr, character sort,
     external selctg, character sense, integer n, complex, dimension( lda, *
     ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer
     sdim, complex, dimension( * ) alpha, complex, dimension( * ) beta,
     complex, dimension( ldvsl, * ) vsl, integer ldvsl, complex, dimension(
     ldvsr, * ) vsr, integer ldvsr, real, dimension( 2 ) rconde, real,
     dimension( 2 ) rcondv, complex, dimension( * ) work, integer lwork,
     real, dimension( * ) rwork, integer, dimension( * ) iwork, integer
     liwork, logical, dimension( * ) bwork, integer info)

CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices.

🎀 Purpose 🎀

Okay so here’s the whole point, bestie: CGGESX computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the complex Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization:

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

where (VSR)**H is the conjugate-transpose of VSR.

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces).

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 is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or for both being zero.

A pair of matrices (S,T) is in generalized complex Schur form if T is upper triangular with non-negative diagonal and S is upper triangular.

🌸 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. It 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 two COMPLEX 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. Note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case INFO is set to N+3 — see INFO below.
  • SENSE — SENSE is CHARACTER*1. It determines which reciprocal condition numbers are computed. = 'N': none are computed; = 'E': computed for average of selected eigenvalues only; = 'V': computed for selected deflating subspaces only; = 'B': computed for both. If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. No exceptions, bestie.
  • N — N is INTEGER. The order of the matrices A, B, VSL, and VSR. N >= 0.
  • A — A is COMPLEX 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 COMPLEX 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.
  • ALPHA — ALPHA is COMPLEX array, dimension (N).
  • BETA — BETA is COMPLEX array, dimension (N). On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j) and BETA(j), j=1,...,N are the diagonals of the complex Schur form (S,T). BETA(j) will be non-negative real.
Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA 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 COMPLEX 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 COMPLEX 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.
  • RCONDE — RCONDE is REAL array, dimension ( 2 ). If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.
  • RCONDV — RCONDV is REAL array, dimension ( 2 ). If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition number for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'.
  • WORK — WORK is COMPLEX 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 if SENSE = 'E', 'V', or 'B', LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may not be large enough.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.
  • RWORK — RWORK is REAL array, dimension ( 8*N ). Real workspace.
  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

LIWORK

(and this is exactly where part 1 of the page cuts off — the LIWORK details roll in with part 2, bestie 💖)

  • LIWORK — she's an INTEGER, and the page says she's the dimension of the array WORK (yes, literally WORK — copying the manpage exactly, no cap 💅). If SENSE = 'N' or N = 0, then LIWORK >= 1; otherwise LIWORK >= N+2.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

  • BWORK — BWORK is a LOGICAL array, dimension (N). Bestie, it's not referenced if SORT = 'N' 🫶.
  • INFO — INFO is INTEGER, and here's the whole mood:
    • = 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 ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
    • > N:
      • =N+1: other than QZ iteration failed in CHGEQZ
      • =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 CTGSEN.

🫶 Author 🫶

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

📍 Definition at line 326 of file cggesx.f

✨ subroutine dggesx ✨

subroutine dggesx (character jobvsl, character jobvsr, character sort,
    external selctg, character sense, integer n, double precision,
    dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *
    ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar,
    double precision, dimension( * ) alphai, double precision, dimension( *
    ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl,
    double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double
    precision, dimension( 2 ) rconde, double precision, dimension( 2 )
    rcondv, double precision, dimension( * ) work, integer lwork, integer,
    dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork,
    integer info)

DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 💜

💫 Purpose 💫

DGGESX computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the real Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives 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; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces) 🌸

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 is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or for 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 🦋

🎀 Parameters 🎀

  • JOBVSL — CHARACTER*1, bestie:
    • 'N': do not compute the left Schur vectors;
    • 'V': compute the left Schur vectors.
  • JOBVSR — CHARACTER*1, same energy on the right side:
    • 'N': do not compute the right Schur vectors;
    • 'V': compute the right Schur vectors.
  • SORT — 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 — a LOGICAL FUNCTION of three DOUBLE PRECISION 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 a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case INFO is set to N+3 😭
  • SENSE — CHARACTER*1. Determines which reciprocal condition numbers are computed:
    • 'N': none are computed;
    • 'E': computed for average of selected eigenvalues only;
    • 'V': computed for selected deflating subspaces only;
    • 'B': computed for both.

If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

  • N — INTEGER. The order of the matrices A, B, VSL, and VSR. N >= 0.
  • A — DOUBLE PRECISION 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 — INTEGER. The leading dimension of A. LDA >= max(1,N).
  • B — DOUBLE PRECISION 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 — INTEGER. The leading dimension of B. LDB >= max(1,N).
  • SDIM — 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 — DOUBLE PRECISION array, dimension (N).
  • ALPHAI — DOUBLE PRECISION array, dimension (N).
  • BETA — DOUBLE PRECISION 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 — DOUBLE PRECISION array, dimension (LDVSL,N). If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'.
  • LDVSL — INTEGER. The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', LDVSL >= N.
  • VSR — DOUBLE PRECISION array, dimension (LDVSR,N). If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'.
  • LDVSR — INTEGER. The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.
  • RCONDE — DOUBLE PRECISION array, dimension ( 2 ). If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.
  • RCONDV — DOUBLE PRECISION array, dimension ( 2 ). If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition numbers for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'.
  • WORK
  • WORK — a DOUBLE PRECISION array, dimension (MAX(1,LWORK)). Bestie, on exit, if INFO = 0, then WORK(1) hands you back the optimal LWORK 👑 (it's like a little receipt telling you what size you should have used).
  • LWORK — INTEGER, the dimension of the WORK array. Here's the sizing tea 🍵: if N = 0, then LWORK >= 1; else if SENSE = 'E', 'V', or 'B', then LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ); otherwise LWORK >= max( 8*N, 6*N+16 ). Also note that 2*SDIM*(N-SDIM) <= N*N/2, and note also that an error is only returned if LWORK < max( 8*N, 6*N+16) — but if SENSE = 'E' or 'V' or 'B', that might legitimately not be large enough 😭. And if LWORK = -1, it's a workspace query: the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA ✨.
  • IWORK — an INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK 🫶.
  • LIWORK — INTEGER, the dimension of the IWORK array. If SENSE = 'N' or N = 0, then LIWORK >= 1; otherwise LIWORK >= N+6.

And if LIWORK = -1, then a workspace query is assumed 🌀: the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

  • BWORK — a LOGICAL array, dimension (N). Not referenced if SORT = 'N' 🙈 (she simply doesn't get invited to that party).
  • INFO — INTEGER, the vibe check at the end 💀:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value, no cap.
    • = 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: something other than QZ iteration failed in DHGEQZ
      • =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 DTGSEN

💌 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧠 Further Details:

An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

     EPS * norm((A, B)) / RCONDE( 1 ).

An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

     EPS * norm((A, B)) / RCONDV( 2 ).

See LAPACK User's Guide, section 4.11 for more information 📚.

Definition at line 361 of file dggesx.f.

🦋 subroutine sggesx

Okay bestie, deep breath, here comes the real-valued sibling and her signature is a whole mood 🎀:

subroutine sggesx (character jobvsl, character jobvsr, character sort,
    external selctg, character sense, 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( 2 )
    rconde, real, dimension( 2 ) rcondv, real, dimension( * ) work, integer
    lwork, integer, dimension( * ) iwork, integer liwork, logical,
    dimension( * ) bwork, integer info)
SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices 💅

✨ Purpose:

SGGESX computes, for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the real Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives 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; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV) 👑. The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces).

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's a reasonable interpretation for beta=0 or for 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. The 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 ✨.

💅 Parameters

  • JOBVSL — CHARACTER*1.
    • = 'N': do not compute the left Schur vectors;
    • = 'V': compute the left Schur vectors.
  • JOBVSR — CHARACTER*1.
    • = 'N': do not compute the right Schur vectors;
    • = 'V': compute the right Schur vectors.
  • SORT — 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 — 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 a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case INFO is set to N+3 😭.

  • SENSE — CHARACTER*1. Determines which reciprocal condition numbers are computed.
    • = 'N': none are computed;
    • = 'E': computed for average of selected eigenvalues only;
    • = 'V': computed for selected deflating subspaces only;
    • = 'B': computed for both.

If SENSE = 'E', 'V', or 'B', SORT must equal 'S' ⚠️.

  • N — INTEGER. The order of the matrices A, B, VSL, and VSR. N >= 0.
  • A — 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 — INTEGER. The leading dimension of A. LDA >= max(1,N).
  • B — 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 — INTEGER. The leading dimension of B. LDB >= max(1,N).
  • SDIM — 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 — REAL array, dimension (N).
  • ALPHAI — REAL array, dimension (N).
  • BETA — ✨ (hold that thought, bestie — this one's deets roll in next)
  • 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.
⚠️ bestie pls read this one: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. So like... do not naively compute the ratio, okay? 🙈 However, ALPHAR and ALPHAI will always be less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).

VSL

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

VSR

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

RCONDE

  • RCONDE — RCONDE is REAL array, dimension ( 2 ). If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'. 🌸

RCONDV

  • RCONDV — RCONDV is REAL array, dimension ( 2 ). If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition numbers for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'. ✨

WORK

  • WORK — WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 💖

LWORK

  • LWORK — LWORK is INTEGER. The dimension of the array WORK. If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B', LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else LWORK >= max( 8*N, 6*N+16 ). Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B' this may not be large enough.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. 👑

IWORK

  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

LIWORK

  • LIWORK — LIWORK is INTEGER. The dimension of the array IWORK. If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise LIWORK >= N+6.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA. 🫶

BWORK

  • BWORK — BWORK is LOGICAL array, dimension (N). Not referenced if SORT = 'N'. 🎀

INFO

  • 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 SHGEQZ
      • =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.

🔥 Further Details:

An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

EPS * norm((A, B)) / RCONDE( 1 ).

An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

EPS * norm((A, B)) / RCONDV( 2 ).

See LAPACK User's Guide, section 4.11 for more information. 📚

Definition at line 361 of file sggesx.f.

🧚 subroutine zggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer sdim, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldvsl, * ) vsl, integer ldvsl, complex*16, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( 2 ) rconde, double precision, dimension( 2 ) rcondv, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

subroutine zggesx (character jobvsl, character jobvsr, character sort,
    external selctg, character sense, integer n, complex*16, dimension( lda,
    * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb,
    integer sdim, complex*16, dimension( * ) alpha, complex*16, dimension( *
    ) beta, complex*16, dimension( ldvsl, * ) vsl, integer ldvsl,
    complex*16, dimension( ldvsr, * ) vsr, integer ldvsr, double precision,
    dimension( 2 ) rconde, double precision, dimension( 2 ) rcondv,
    complex*16, dimension( * ) work, integer lwork, double precision,
    dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork,
    logical, dimension( * ) bwork, integer info)

ZGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices ✨

🌈 Purpose:

ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the complex Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization

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

where (VSR)**H is the conjugate-transpose of VSR. 🪄

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper triangular matrix S and the upper triangular matrix T; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces). 🦋

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 is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or for both being zero. 👀

A pair of matrices (S,T) is in generalized complex Schur form if T is upper triangular with non-negative diagonal and S is upper triangular. slay. 💅

💜 Parameters

JOBVSL

  • JOBVSL — JOBVSL is CHARACTER*1
    • = 'N': do not compute the left Schur vectors;
    • = 'V': compute the left Schur vectors.

JOBVSR

  • JOBVSR — JOBVSR is CHARACTER*1
    • = 'N': do not compute the right Schur vectors;
    • = 'V': compute the right Schur vectors.

SORT

  • 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 — SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 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.
👀 heads up: a selected complex eigenvalue may no longer satisfy SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned), in this case INFO is set to N+3 — see INFO below. not me warning you twice 💀

SENSE

  • SENSE — SENSE is CHARACTER*1. Determines which reciprocal condition numbers are computed.
    • = 'N': None are computed;
    • = 'E': Computed for average of selected eigenvalues only;
    • = 'V': Computed for selected deflating subspaces only;
    • = 'B': Computed for both.

If SENSE = 'E', 'V', or 'B', SORT must equal 'S'. 🚨

N

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

A

  • A — A is COMPLEX*16 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, bestie. It's the leading dimension of A, and the rule is LDA >= max(1,N). No cap. ✨
  • B — B is COMPLEX*16 array, dimension (LDB, N). On entry, it's the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T. She's booked and busy. 💅
  • LDB — LDB is INTEGER. It's the leading dimension of B, and LDB >= max(1,N). Respect the leading dimension or catch hands. 🫶
  • SDIM — SDIM is INTEGER. If SORT = 'N', then SDIM = 0. If SORT = 'S', then SDIM = number of eigenvalues (after sorting) for which SELCTG is true.
  • ALPHA — ALPHA is COMPLEX*16 array, dimension (N). That's it, that's the whole entry.
  • BETA — BETA is COMPLEX*16 array, dimension (N). On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T). BETA(j) will be non-negative real.
Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). So like… don't just divide and pray, okay? 😭
  • VSL — VSL is COMPLEX*16 array, dimension (LDVSL,N). If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'. Left Schur vectors stay iconic. 👑
  • LDVSL — LDVSL is INTEGER. It's the leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL = 'V', then LDVSL >= N.
  • VSR — VSR is COMPLEX*16 array, dimension (LDVSR,N). If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'. Right Schur vectors, main character energy. 🌟
  • LDVSR — LDVSR is INTEGER. It's the leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', then LDVSR >= N.
  • RCONDE — RCONDE is DOUBLE PRECISION array, dimension ( 2 ). If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'. Condition number tea, served hot. 🍓
  • RCONDV — RCONDV is DOUBLE PRECISION array, dimension ( 2 ). If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition number for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'.
  • WORK — WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. Workspace said "I'll handle it" and honestly she did. ✨
  • LWORK — LWORK is INTEGER. It's the dimension of the array WORK.

If N = 0, then LWORK >= 1; else if SENSE = 'E', 'V', or 'B', then LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)); else LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an error is only returned if LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may not be large enough.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

  • RWORK — RWORK is DOUBLE PRECISION array, dimension ( 8*N ). Real workspace. Keeping it real, literally. 💜
  • IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
  • LIWORK — LIWORK is INTEGER. It's the dimension of the array IWORK.

If SENSE = 'N' or N = 0, then LIWORK >= 1, otherwise LIWORK >= N+2.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the bound on the optimal size of the WORK array and the minimum size of the IWORK array, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

  • BWORK — BWORK is LOGICAL array, dimension (N). Not referenced if SORT = 'N'. Logic only when needed, periodt. 💅
  • INFO — INFO is INTEGER. Here's the whole status tea:
    • = 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 ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.
    • > N:
      • =N+1: other than QZ iteration failed in ZHGEQZ
      • =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 ZTGSEN.

Author 🌸

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

Definition at line 326 of file zggesx.f.

Author ✨

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