SRC/zgebal.f(3)

🎀 NAME 🎀

SRC/zgebal.f — yes bestie, that's literally the file, and this whole page is its main character moment in the Library Functions Manual. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

       subroutine zgebal (job, n, a, lda, ilo, ihi, scale, info)
           ZGEBAL

💅 Function/Subroutine Documentation 💅

The icon herself:

   subroutine zgebal (character job, integer n, complex*16, dimension( lda, * )
       a, integer lda, integer ilo, integer ihi, double precision, dimension( *
       ) scale, integer info)
       ZGEBAL

🎯 Purpose 🎯

Okay so ZGEBAL balances a general complex matrix A. This involves, first, permuting 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, 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 — choose your fighter. 🫶

Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors. Periodt. 🌈

🧚 Parameters 🧚

• JOB is CHARACTER*1 and 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 is INTEGER — the order of the matrix A. N >= 0. ✨
• 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. See Further Details below. 👀
• LDA is INTEGER — the leading dimension of the array A. LDA >= max(1,N). 💅
• ILO is INTEGER. That's the whole lore, bestie. 🎀
• 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 is DOUBLE PRECISION 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 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 CBAL.

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 164 of file zgebal.f 📍

Definition at line 164 of file zgebal.f.

🛠️ Author 🛠️

Generated automatically by Doxygen for LAPACK from the source code.

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💖 Later, bestie! This whole page is from LAPACK Version 3.12.0, filed as SRC/zgebal.f(3) — and yes, she ate. 🎀✨