SRC/cpbtrf.f(3)

✨ CPBTRF — the Cholesky glow-up for band matrices ✨

okay so bestie, you know how SRC/cpbtrf.f is a Fortran source file in LAPACK, right? well this page is the manual for the little icon inside it: a subroutine called cpbtrf. it's giving "linear algebra but make it cute", and honestly? it ate. 💅

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🌈 SYNOPSIS 🌈

Functions/Subroutines

• subroutine cpbtrf (uplo, n, kd, ab, ldab, info) — the whole vibe. lowercase name, big responsibilities.

subroutine cpbtrf (character uplo, integer n, integer kd, complex,
     dimension( ldab, * ) ab, integer ldab, integer info)

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🦋 Function/Subroutine Documentation 🦋

💖 subroutine cpbtrf (character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, integer info)

CPBTRF

Purpose:

lowkey the main character moment: CPBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. no cap. 🫶

The factorization has the form

A = U**H * U,  if UPLO = 'U', or
A = L  * L**H,  if UPLO = 'L',

where U is an upper triangular matrix and L is lower triangular. so either way you get your triangles, it's just a question of which side is serving. 🔺

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Parameters

UPLO

• UPLO is CHARACTER*1
  • = 'U': Upper triangle of A is stored; ⬆️
  • = 'L': Lower triangle of A is stored. ⬇️

N

• N is INTEGER
  • The order of the matrix A. N >= 0.

KD

• KD is INTEGER
  • The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.

AB

• AB is COMPLEX array, dimension (LDAB,N)
  • On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows:
    • if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    • if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
  • On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, in the same storage format as A. she leaves the party in the exact same outfit, iconic. 🎀

LDAB

• LDAB is INTEGER
  • The leading dimension of the array AB. LDAB >= KD+1.

INFO

• INFO is INTEGER
  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value 😭
  • > 0: if INFO = i, the leading principal minor of order i is not positive, and the factorization could not be completed. 💀 that's the "not positive definite" heartbreak, fr fr.

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Author

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

(the whole group chat contributed, we love a collab 💜)

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Further Details:

okay this part is genuinely iconic, the band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U':

On entry: On exit:

    *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry: On exit:

   a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
   a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
   a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

Array elements marked * are not used by the routine. they're literally just there for the aesthetic, do not touch them, they are not a part of this story. ✨

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Contributors:

Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989. shoutout to the OGs, 1989 and still slaying 🕰️👑

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Definition at line 141 of file cpbtrf.f.

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👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code. not a human wrote this bit, it's the machine's little bow on top 🎀

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and that's the page, bestie! this whole thing came from LAPACK, Version 3.12.0, and the source file was SRC/cpbtrf.f(3) — so if you ever need to cite it, you know exactly where it lives. mwah, go factor some Hermitian positive definite band matrices responsibly 💋🌸