SRC/dgebal.f(3)

Bestie, welcome to SRC/dgebal.f(3), straight out of the Library Functions Manual ✨. This is the whole tea on DGEBAL, no cap.

💖 NAME 💖

SRC/dgebal.f — yep, that’s the file, and it’s serving LAPACK function-documentation realness 🫶.

✨ SYNOPSIS ✨

Functions/Subroutines
    subroutine dgebal (job, n, a, lda, ilo, ihi, scale, info)
        DGEBAL

🦋 Function/Subroutine Documentation 🦋

subroutine dgebal (character job, integer n, double precision, dimension(
    lda, * ) a, integer lda, integer ilo, integer ihi, double precision,
    dimension( * ) scale, integer info)

DGEBAL — iconic subroutine behavior incoming 👑

🎀 Purpose 🎀

Okay so DGEBAL balances a general real matrix A. It does this in two steps, and both steps are optional:

1. It permutes A by a similarity transformation, isolating eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal.
2. It applies a diagonal similarity transformation to rows and columns ILO to IHI, making the rows and columns as close in norm as possible.

Balancing may reduce the 1-norm of the matrix, and it may improve the accuracy of the computed eigenvalues and/or eigenvectors. Slay for numerical stability 💅.

💅 Parameters 💅

JOB

JOB is CHARACTER*1. It specifies the operations to be performed on A:

• = 'N': none: simply set ILO = 1, IHI = N, SCALE(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 matrix A. 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.
• See Further Details.

LDA

LDA is INTEGER. The leading dimension of the array A. LDA >= 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 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.

If JOB = 'N' or 'S', ILO = 1 and IHI = N.

SCALE

SCALE is DOUBLE PRECISION array, dimension (N). It holds details of the permutations and scaling factors applied to A.

If P(j) is the index of the row and column interchanged with row and column j, and D(j) is the scaling factor applied to row and column j, then:

SCALE(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.

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 🔥

The permutations consist of row and column interchanges which put the matrix in the form

            ( T1   X   Y  )
    P A P = (  0   B   Z  )
            (  0   0   T2 )

where T1 and T2 are upper triangular matrices whose eigenvalues lie along the diagonal. The column indices ILO and IHI mark the starting and ending columns of the submatrix B.

Balancing consists of applying a diagonal similarity transformation inv(D) * B * D to make the 1-norms of each row of B and its corresponding column nearly equal. The output matrix is

    ( T1     X*D          Y    )
    (  0  inv(D)*B*D  inv(D)*Z ).
    (  0      0           T2   )

Information about the permutations P and the diagonal matrix D is returned in the vector SCALE.

This subroutine is based on the EISPACK routine BALANC.

Modified by Tzu-Yi Chen, Computer Science Division, University of California at Berkeley, USA.

Refactored by Evert Provoost, Department of Computer Science, KU Leuven, Belgium.

Definition at line 162 of file dgebal.f.

🧚 Author 🧚

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And that’s the cute little sign-off, bestie 💜: this page is from LAPACK, Version 3.12.0, file SRC/dgebal.f(3). Bye and stay balanced 🫶✨