ZGEBAL(3)

🦋 ZGEBAL(3) — the zgebal.f(3) glow-up ✨

okay so real talk, this page lives at zgebal.f in the LAPACK manual (manual name: zgebal.f(3)), and it's the whole lore drop on ZGEBAL. it's giving "linear algebra but make it a makeover montage" 💅. also, prose warning: this manpage is technically about a file path, the synopsis is literally the path, and it's iconic for that. no cap.

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zgebal.f — that's it, that's the whole name bestie, it's the file, she's the file 🎀

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🌸 SYNOPSIS 🌸

Functions/Subroutines 🫶

• subroutine ZGEBAL (job, n, a, lda, ilo, ihi, scale, info) — iconic, no notes, this is the one

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💖 Function/Subroutine Documentation 💖

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:

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

translation: she balances your general complex matrix A, and it's a two-part serve 💅. step one, permute 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. step two, apply a diagonal similarity transformation to rows and columns ILO to IHI so the rows and columns are as close in norm as possible. and the best part? both steps are optional — she's flexible, she's lowkey chill 😌

why bother? because balancing may reduce the 1-norm of the matrix and improve the accuracy of the computed eigenvalues and/or eigenvectors. it's a glow-up for your numbers, fr fr ✨

🍓 Parameters

• JOB — JOB is CHARACTER*1. specifies the operations to be performed on A, and she has range:
  • = '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. non-negotiable bestie.
• 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. see Further Details 💜 (she get a whole section, main character energy)
• LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,N). respect the bound 🙏
• ILO — ILO is INTEGER. (yes that's the whole description, she's a mystery, she's reserved 🌈)
• 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. (note the capital I in I = IHI+1,...,N — that's how she wrote it, we don't touch it 💀)
• SCALE — 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. order matters bestie, don't freestyle it 🎀

• INFO — INFO is INTEGER:
  • = 0: successful exit. slay 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value. not slay 😭

👑 Author

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

a whole collective, a whole squad 🫶

⭐ 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 !>

so the permutations are row-and-column interchanges that put the matrix in this exact shape:

            ( 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 is 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 — matching energy, we love to see it 💅. and 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 — she keeps receipts, iykyk 🧾✨

📌 credit where it's due: 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. legends only 👑

🌈 Definition at line 164 of file zgebal.f.

that's the coordinate bestie, line 164 of zgebal.f, go look 👀

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🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. yes she was built by a robot and she still ate 🤖💖

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okay bye bestie, go balance that matrix 🦋💜

LAPACK — Version 3.12.0 — and that path again, for the record: /home/abui...RC/zgebal.f(3) (yes, abbreviated, that's how the footer rolls). mwah, love u, see u in the next manpage 😘🌷