SRC/sgebal.f(3)

💖 NAME 💖

okay so this is SRC/sgebal.f(3) from the Library Functions Manual, bestie ✨ It's the page for the LAPACK subroutine sgebal, and it's giving “balance a general real matrix” energy.

SRC/sgebal.f — that's the page name, periodt.

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✨ SYNOPSIS ✨

Functions/Subroutines

   Functions/Subroutines
       subroutine sgebal (job, n, a, lda, ilo, ihi, scale, info)
           SGEBAL

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🦋 Function/Subroutine Documentation 🦋

Here's the signature, copied exactly because we do not play with literals:

   subroutine sgebal (character job, integer n, real, dimension( lda, * ) a,
       integer lda, integer ilo, integer ihi, real, dimension( * ) scale,
       integer info)
       SGEBAL

SGEBAL is the main character here 👑

🎯 Purpose 🎯

SGEBAL balances a general real matrix A. The way that works is: first, it permutes A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and 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, no cap.

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

🎀 Parameters 🎀

• JOB — JOB is CHARACTER*1. 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 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. 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 REAL array, dimension (N). 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 sgebal.f.

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🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

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okay byeeee 💖 This footer is giving LAPACK Version 3.12.0 and SRC/sgebal.f(3); the date isn't listed on the page, so I'm not inventing one, bestie 🫶🎀