SRC/zggbal.f(3)

NAME 🌸

SRC/zggbal.f

Hey bestie 💖 welcome to the Library Functions Manual page for SRC/zggbal.f — this is the whole vibe, and the star of the show is ZGGBAL. No cap, we’re keeping every single detail.

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SYNOPSIS ✨

Functions/Subroutines 🎀

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

ZGGBAL

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Function/Subroutine Documentation 🫶

ZGGBAL 👑

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 🎯

Okay so 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.

Parameters 📋

• JOB — JOB is CHARACTER*1. It 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*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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• B — B is 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 — 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

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 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 zggbal.f.

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Author 💅

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