SRC/cgbequ.f(3)

✨ SRC/cgbequ.f ✨

okay so bestie, welcome to the glow-up era of SRC/cgbequ.f — this is the LAPACK page for cgbequ, and it's genuinely one of those little routines that quietly does the most. 💅

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🎀 SYNOPSIS 🎀

Functions/Subroutines

• subroutine cgbequ (m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info) — the vibe check of the whole page, this is CGBEQU herself 👑

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💖 Function/Subroutine Documentation 💖

🦋 subroutine cgbequ (integer m, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) r, real, dimension( * ) c, real rowcnd, real colcnd, real amax, integer info)

CGBEQU

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🌸 Purpose

!>
!> CGBEQU computes row and column scalings intended to equilibrate an
!> M-by-N band matrix A and reduce its condition number.  R returns the
!> row scale factors and C the column scale factors, chosen to try to
!> make the largest element in each row and column of the matrix B with
!> elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
!>
!> R(i) and C(j) are restricted to be between SMLNUM = smallest safe
!> number and BIGNUM = largest safe number.  Use of these scaling
!> factors is not guaranteed to reduce the condition number of A but
!> works well in practice.
!>

so like, CGBEQU is out here computing row and column scalings to equilibrate an M-by-N band matrix A and bring its condition number down. R comes back with the row scale factors, C with the column scale factors — picked so the biggest element in each row and column of the matrix B (that's the one where B(i,j)=R(i)*A(i,j)*C(j), keep up bestie) has absolute value 1. iconic behaviour tbh. 🫶

R(i) and C(j) are locked between SMLNUM = smallest safe number and BIGNUM = largest safe number. and real talk, using these scaling factors is not guaranteed to reduce the condition number of A, but it works well in practice — no cap, that's straight from the page. 💯

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💜 Parameters

• M — is INTEGER. The number of rows of the matrix A. M >= 0. 📏
• N — is INTEGER. The number of columns of the matrix A. N >= 0.
• KL — is INTEGER. The number of subdiagonals within the band of A. KL >= 0. (the down-bad diagonals ✨)
• KU — is INTEGER. The number of superdiagonals within the band of A. KU >= 0. (the up-and-over diagonals 🦋)
• AB — is COMPLEX array, dimension (LDAB,N). The band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB like this:

AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).

• LDAB — is INTEGER. The leading dimension of the array AB. LDAB >= KL+KU+1. 👑
• R — is REAL array, dimension (M). If INFO = 0, or INFO > M, R contains the row scale factors for A. 🌈
• C — is REAL array, dimension (N). If INFO = 0, C contains the column scale factors for A. ⭐
• ROWCND — is REAL. If INFO = 0 or INFO > M, ROWCND contains the ratio of the smallest R(i) to the largest R(i). If ROWCND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by R. 💀
• COLCND — is REAL. If INFO = 0, COLCND contains the ratio of the smallest C(i) to the largest C(i). If COLCND >= 0.1, it is not worth scaling by C.
• AMAX — is REAL. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. 🔥
• INFO — is INTEGER:
  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value 🙈
  • > 0: if INFO = i, and i is
    • <= M: the i-th row of A is exactly zero
    • > M: the (i-M)-th column of A is exactly zero

lowkey important: those zero row / zero column situations are real and they come back to you in INFO, so don't sleep on checking it. 😭

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🧚 Author

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

Definition at line 152 of file cgbequ.f. 📍

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🍓 Author

Generated automatically by Doxygen for LAPACK from the source code. (yes bestie, the robots wrote the page, the humans wrote the Fortran 💫)

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LAPACK Version 3.12.0 — that's the whole mood, mwah, ily, go equilibrate something 💖🎀🦋✨