SRC/zpbtf2.f(3)

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💖 NAME 💖

SRC/zpbtf2.f

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✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

• subroutine zpbtf2 (uplo, n, kd, ab, ldab, info) — ZPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix using the unblocked algorithm. slay.

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🎀 Function/Subroutine Documentation 🎀

💅 subroutine zpbtf2 💅

subroutine zpbtf2 (character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, integer info)

ZPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix, unblocked-algorithm style ✨

Purpose

okay so listen: ZPBTF2 computes the Cholesky factorization of a complex Hermitian positive definite band matrix A 💅

the factorization has this form:

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

where U is an upper triangular matrix, U**H is the conjugate transpose of U, and L is lower triangular 🦋

this is the unblocked version of the algorithm, calling Level 2 BLAS — lowkey iconic, no cap.

Parameters

• UPLO — is CHARACTER*1. it specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = 'U': Upper triangular
  • = 'L': Lower triangular
• N — is INTEGER. the order of the matrix A. constraint: N >= 0 💖
• KD — is INTEGER. the number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. constraint: KD >= 0 🌸
• AB — is COMPLEX*16 array, dimension (LDAB,N). on entry, it holds 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 like this:
  • 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, it holds 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 👑

• LDAB — is INTEGER. the leading dimension of the array AB. constraint: LDAB >= KD+1 ✨
• INFO — is INTEGER:
  • = 0: successful exit 🎉
  • < 0: if INFO = -k, the k-th argument had an illegal value 🙈
  • > 0: if INFO = k, the leading principal minor of order k is not positive, and the factorization could not be completed. so yeah, that’s a real “oop” moment 💀

Author

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

💜 Further Details: 💜

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 just there vibing, honestly 🫶

Definition at line 141 of file zpbtf2.f.

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🌈 Author 🌈

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