SRC/sggbal.f(3)

💅 SRC/sggbal.f(3)

okay so bestie, this page is from the Library Functions Manual, and it’s all about SRC/sggbal.f — the file, the icon, the main character. 💖

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✨ NAME ✨

SRC/sggbal.f — that’s the file, no cap. 🎀

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🌈 SYNOPSIS 🌈

🧚 Functions/Subroutines

subroutine sggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale,
    work, info)

SGGBAL

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🦋 Function/Subroutine Documentation 🦋

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

okay so SGGBAL balances a pair of general real 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. That’s the whole vibe check. 💅

🎀 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 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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• B — B is 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 — 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:

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.

yes, the source really keeps LSCALE(j) in this RSCALE formula block — I’m preserving it exactly, bestie 😭💀

• 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 sggbal.f.

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⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code.

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and that’s the cute little footer stamp, bestie: LAPACK Version 3.12.0, from SRC/sggbal.f(3). periodt. 💖🫶✨