DGGBAL(3)

💅 NAME

okay so bestie, this is dggbal.f(3) from the Library Functions Manual — the header path situation is /home/abui...RC/dggbal.f(3) (yes, it appears twice, she’s extra 💅), and the actual source file path is:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dggbal.f

that’s the file behind this manual page, periodt.

✨ SYNOPSIS

here’s the call preview, exactly as written:

   Functions/Subroutines
       subroutine DGGBAL (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale,
           work, info)
           DGGBAL

🫶 Function/Subroutine Documentation

the full signature, character-for-character:

   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

DGGBAL is the bestie that 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. second, 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 — choose your adventure with JOB, no cap.

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. iconic behavior honestly.

🌸 Parameters

okay parameter roll call — every single one, no skips:

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 — see IHI right below, they’re a package deal 🫶

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) — the tea on 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) — same energy, but for 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.

yes, the source literally writes LSCALE(j) in the RSCALE block too — I’m preserving that exactly, no autocorrect energy 💀. 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

the credited institutions are:

• 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.

🦋 Author (generated)

Generated automatically by Doxygen for LAPACK from the source code.

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okay that’s the whole page, bestie. the footer source/version line is LAPACK Version 3.12.0 /home/abui...RC/dggbal.f(3) — the footer has no date, so I’m not inventing one 💀. go balance those generalized eigenvalue problems and slay 💅✨