SRC/zpbsv.f(3)

📦
version 3.12.1 · 7 versions
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💖 NAME 💖

This is SRC/zpbsv.f(3) from the Library Functions Manual — manual name folded in, bestie, don't worry 💅 — and the NAME on the page is SRC/zpbsv.f. It's giving “LAPACK routine with main character energy” fr fr. 🎀

✨ 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

💅 Function/Subroutine Documentation 💅

🦋 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)

Lowkey the headline is: ZPBSV computes the solution to system of linear equations A * X = B for OTHER matrices — yes, capital-OTHER matrices, she's not like the other girls 💅✨

🍓 Purpose: 🍓

Okay so bestie, 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. Iconic.

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. Periodt. 💖

🌸 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.

👑 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. Not invited to the function, honestly 💀

🫶 Definition at line 163 of file zpbsv.f 🫶

Definition at line 163 of file zpbsv.f.

🧚 Author 🧚

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