ggbal(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🎀 NAME 🎀

ggbal — balance matrix

✨ SYNOPSIS ✨

Functions

  • subroutine cggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info) — CGGBAL
  • subroutine dggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info) — DGGBAL
  • subroutine sggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info) — SGGBAL
  • subroutine zggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info) — ZGGBAL

💅 Detailed Description 💅

okay so this page is literally the glow-up guide for balancing a pair of general matrices — the routines are cggbal, dggbal, sggbal, and zggbal, and they live in the Function Documentation section below. ✨

🌸 Function Documentation 🌸

🦋 subroutine cggbal (character job, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real, dimension( * ) work, integer info) 🦋

CGGBAL

Purpose

okay so CGGBAL balances a pair of general complex matrices (A,B). First it permutes A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; then it applies a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional, bestie. ✨ Balancing may reduce the 1-norm of the matrices and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. 💖

Parameters

  • JOB — JOB is CHARACTER*1. Specifies the operations to be performed on A and B:
    • = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and RSCALE(I) = 1.0 for i=1,...,N;
    • = 'P': permute only;
    • = 'S': scale only;
    • = 'B': both permute and scale.
  • 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 input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = 'N', A is not referenced.
  • 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 input matrix B. On exit, B is overwritten by the balanced matrix. If JOB = 'N', B is not referenced.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • ILO — ILO is INTEGER.
  • IHI — IHI is INTEGER. ILO and IHI are set to integers such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N.
  • LSCALE — LSCALE is REAL array, dimension (N). Details of the permutations and scaling factors applied to the left side of A and B. If P(j) is the index of the row interchanged with row j, and D(j) is the scaling factor applied to row j, then:
LSCALE(j) = P(j)    for J = 1,...,ILO-1
          = D(j)    for J = ILO,...,IHI
          = P(j)    for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

  • RSCALE — RSCALE is REAL array, dimension (N). Details of the permutations and scaling factors applied to the right side of A and B. If P(j) is the index of the column interchanged with column j, and D(j) is the scaling factor applied to column j, then:
RSCALE(j) = P(j)    for J = 1,...,ILO-1
          = D(j)    for J = ILO,...,IHI
          = P(j)    for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

  • WORK — WORK is REAL array, dimension (lwork). lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at least 1 when JOB = 'N' or 'P'.
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author

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

Further Details

See R.C. WARD, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 175 of file cggbal.f.

🦋 subroutine dggbal (character job, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, double precision, dimension( * ) lscale, double precision, dimension( * ) rscale, double precision, dimension( * ) work, integer info) 🦋

DGGBAL

Purpose

okay so DGGBAL balances a pair of general real matrices (A,B). First it permutes A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; then it applies a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional, bestie. ✨ Balancing may reduce the 1-norm of the matrices and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. 💖

Parameters

  • JOB — JOB is CHARACTER*1. Specifies the operations to be performed on A and B:
    • = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and RSCALE(I) = 1.0 for i = 1,...,N.
    • = 'P': permute only;
    • = 'S': scale only;
    • = 'B': both permute and scale.
  • N — N is INTEGER. The order of the matrices A and B. N >= 0.
  • A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = 'N', A is not referenced.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is DOUBLE PRECISION array, dimension (LDB,N). On entry, the input matrix B. On exit, B is overwritten by the balanced matrix. If JOB = 'N', B is not referenced.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • ILO — ILO is INTEGER.
  • IHI — IHI is INTEGER. ILO and IHI are set to integers such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N.
  • LSCALE — LSCALE is DOUBLE PRECISION array, dimension (N). Details of the permutations and scaling factors applied to the left side of A and B. If P(j) is the index of the row interchanged with row j, and D(j) is the scaling factor applied to row j, then:
LSCALE(j) = P(j)    for J = 1,...,ILO-1
          = D(j)    for J = ILO,...,IHI
          = P(j)    for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

  • RSCALE — RSCALE is DOUBLE PRECISION array, dimension (N). Details of the permutations and scaling factors applied to the right side of A and B. If P(j) is the index of the column interchanged with column j, and D(j) is the scaling factor applied to column j, then:
LSCALE(j) = P(j)    for J = 1,...,ILO-1
          = D(j)    for J = ILO,...,IHI
          = P(j)    for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

  • WORK — WORK is DOUBLE PRECISION array, dimension (lwork). lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at least 1 when JOB = 'N' or 'P'.
  • INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author

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

Further Details

See R.C. WARD, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 175 of file dggbal.f.

🦋 subroutine sggbal (character job, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real, dimension( * ) work, integer info) 🦋

SGGBAL

Purpose

(SGGBAL's purpose and parameters continue in part 2 — stay tuned, bestie! 💖)

...and that's the why, bestie. Let me hit you with the payoff: balancing may reduce the 1-norm of the matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. ✨ Like, that's the whole point. That's the glow-up. 💅

💖 Parameters 💖

  • JOB — CHARACTER*1. Specifies the operations to be performed on A and B:
    • = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and RSCALE(I) = 1.0 for i = 1,...,N.
    • = 'P': permute only;
    • = 'S': scale only;
    • = 'B': both permute and scale.
  • N — INTEGER. The order of the matrices A and B. N >= 0. (Yes, zero is allowed, she's chill like that. 🫶)
  • A — REAL array, dimension (LDA,N). On entry, the input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = 'N', A is not referenced. 🚫
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — REAL array, dimension (LDB,N). On entry, the input matrix B. On exit, B is overwritten by the balanced matrix. If JOB = 'N', B is not referenced. 🚫
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • ILO — INTEGER. (she's an output girlie, just vibing until the routine fills her in 🎀)
  • IHI — INTEGER. ILO and IHI are set to integers such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N.
  • LSCALE — REAL array, dimension (N). Details of the permutations and scaling factors applied to the left side of A and B. If P(j) is the index of the row interchanged with row j, and D(j) is the scaling factor applied to row j, then:
  LSCALE(j) = P(j)    for J = 1,...,ILO-1
            = D(j)    for J = ILO,...,IHI
            = P(j)    for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. 🌀 (order matters, don't @ me)

  • RSCALE — REAL array, dimension (N). Details of the permutations and scaling factors applied to the right side of A and B. If P(j) is the index of the column interchanged with column j, and D(j) is the scaling factor applied to column j, then:
  LSCALE(j) = P(j)    for J = 1,...,ILO-1
            = D(j)    for J = ILO,...,IHI
            = P(j)    for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. 🌀

Ok real talk — yes, that second block literally says LSCALE(j) in the docs even though it's the RSCALE section. That's how it's written in the source page, so I'm not touching it. Not my typo, not my problem, we reproduce faithfully in this house. 💀
  • WORK — REAL array, dimension (lwork). lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at least 1 when JOB = 'N' or 'P'. Size your scratch space accordingly, she is not flexible about this. 😤
  • INFO — INTEGER:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value. 🚨

🌸 Author 🌸

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

🦋 Further Details 🦋

See R.C. WARD, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141-152. 📚

Definition at line 175 of file sggbal.f.

👑 ZGGBAL 👑

Ok so now the complex version, and honestly she ate. Same energy, fancy numbers. Here's the signature in all her glory:

subroutine zggbal (character job, integer n, complex*16, dimension( lda, * )
    a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer
    ilo, integer ihi, double precision, dimension( * ) lscale, double
    precision, dimension( * ) rscale, double precision, dimension( * ) work,
    integer info)

✨ Purpose ✨

ZGGBAL balances a pair of general complex matrices (A,B). This involves, first, permuting A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. 🎀

Balancing may reduce the 1-norm of the matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized eigenvalue problem A*x = lambda*B*x. Same deal as her real sister, just with COMPLEX*16 energy. 💜

💖 Parameters 💖

  • JOB — CHARACTER*1. Specifies the operations to be performed on A and B:
    • = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0 and RSCALE(I) = 1.0 for i=1,...,N;
    • = 'P': permute only;
    • = 'S': scale only;
    • = 'B': both permute and scale.
  • N — INTEGER. The order of the matrices A and B. N >= 0.
  • A — COMPLEX*16 array, dimension (LDA,N). On entry, the input matrix A. On exit, A is overwritten by the balanced matrix. If JOB = 'N', A is not referenced.
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — COMPLEX*16 array, dimension (LDB,N). On entry, the input matrix B. On exit, B is overwritten by the balanced matrix. If JOB = 'N', B is not referenced.
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • ILO — INTEGER.
  • IHI — INTEGER. ILO and IHI are set to integers such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If JOB = 'N' or 'S', ILO = 1 and IHI = N.
  • LSCALE — DOUBLE PRECISION array, dimension (N). Details of the permutations and scaling factors applied to the left side of A and B. If P(j) is the index of the row interchanged with row j, and D(j) is the scaling factor applied to row j, then:
  LSCALE(j) = P(j)    for J = 1,...,ILO-1
            = D(j)    for J = ILO,...,IHI
            = P(j)    for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

  • RSCALE — DOUBLE PRECISION array, dimension (N). Details of the permutations and scaling factors applied to the right side of A and B. If P(j) is the index of the column interchanged with column j, and D(j) is the scaling factor applied to column j, then:
  RSCALE(j) = P(j)    for J = 1,...,ILO-1
            = D(j)    for J = ILO,...,IHI
            = P(j)    for J = IHI+1,...,N.

The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1. (And yes, this one says RSCALE(j) properly — the complex girlie got it right. 🌈)

  • WORK — DOUBLE PRECISION array, dimension (lwork). lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and at least 1 when JOB = 'N' or 'P'.
  • INFO — INTEGER:
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value. 🚨

🌸 Author 🌸

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

🦋 Further Details 🦋

See R.C. WARD, Balancing the generalized eigenvalue problem, SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 175 of file zggbal.f.

🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. 🤖 (the robots did the paperwork, we just vibe)

aaand that's the whole page, bestie, you made it to the end!! 🥳💖 This was ggbal(3) from LAPACK, Version 3.12.0 — go balance something iconic, mwah 💅✨🦋

📦 where this page comes from