/home/abui...RC/zpbsv.f(3)

🎀 /home/abui...RC/zpbsv.f(3) 🎀

okay so this is a page from the Library Functions Manual bestie, and it's all about one lil subroutine that is genuinely doing the most 💅

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zpbsv.f

like... that's the whole name. that's the path. she's literally just a file path and honestly? iconic. 👑

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✨ SYNOPSIS ✨

🌸 Functions/Subroutines

subroutine ZPBSV (uplo, n, kd, nrhs, ab, ldab, b, ldb, info)

ZPBSV computes the solution to system of linear equations A * X = B for OTHER matrices ✨

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💖 Function/Subroutine Documentation 💖

🦋 subroutine ZPBSV

subroutine ZPBSV (character uplo, integer n, integer kd, integer nrhs,
     complex*16, dimension( ldab, * ) ab, integer ldab, complex*16,
     dimension( ldb, * ) b, integer ldb, integer info)

ZPBSV computes the solution to system of linear equations A * X = B for OTHER matrices

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🌈 Purpose:

!>
!> ZPBSV computes the solution to a complex system of linear equations
!>    A * X = B,
!> where A is an N-by-N Hermitian positive definite band matrix and X
!> and B are N-by-NRHS matrices.
!>
!> The Cholesky decomposition is used to factor A as
!>    A = U**H * U,  if UPLO = 'U', or
!>    A = L * L**H,  if UPLO = 'L',
!> where U is an upper triangular band matrix, and L is a lower
!> triangular band matrix, with the same number of superdiagonals or
!> subdiagonals as A.  The factored form of A is then used to solve the
!> system of equations A * X = B.
!>

so basically she takes your Hermitian positive definite band matrix A, does a Cholesky glow-up on it (A = U**H * U when UPLO = 'U', or A = L * L**H when UPLO = 'L') — where U is an upper triangular band matrix and L is a lower triangular band matrix, both keeping the same number of superdiagonals or subdiagonals as A — and then uses that factored form to solve A * X = B. slay 💅

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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 number of linear equations, i.e., 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 🌸
• NRHS — NRHS is INTEGER. the number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0 💜
• AB — AB is COMPLEX*16 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).

see below for further details 👀

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 💖

• LDAB — LDAB is INTEGER. the leading dimension of the array AB. LDAB >= KD+1 🎀
• B — B is COMPLEX*16 array, dimension (LDB,NRHS). on entry, the N-by-NRHS right hand side matrix B. on exit, if INFO = 0, the N-by-NRHS solution matrix X ✨
• LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1,N) 🌟
• 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 of A is not positive, so the factorization could not be completed, and the solution has not been computed 💀

ngl if you get a positive INFO, the factorization did NOT finish and your solution does not exist. no cap. don't go reading uninitialised memory bestie 🫶

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🌸 Author 🌸

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

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⭐ 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 🙈 (literally just vibes, they sit there unused)

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🎀 Definition at line 163 of file zpbsv.f.

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💫 Author 💫

Generated automatically by Doxygen for LAPACK from the source code.

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🫶 and that's the whole page bestie!! LAPACK — Version 3.12.0, and the file she lives in is /home/abui...RC/zpbsv.f(3). go forth and factor your band matrices, ily, bye 💅✨🎀