SRC/dpbsv.f(3)

💅 SRC/dpbsv.f(3) — Library Functions Manual 💅

✨ NAME ✨

SRC/dpbsv.f — bestie, this is the manual page for the dpbsv routine.

🦋 SYNOPSIS 🦋

Functions/Subroutines

       subroutine dpbsv (uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
            DPBSV computes the solution to system of linear equations A * X = B
           for OTHER matrices

💖 Function/Subroutine Documentation 💖

   subroutine dpbsv (character uplo, integer n, integer kd, integer nrhs,
       double precision, dimension( ldab, * ) ab, integer ldab, double
       precision, dimension( ldb, * ) b, integer ldb, integer info)

The one-liner is literally: DPBSV computes the solution to system of linear equations A * X = B for OTHER matrices ✨

Purpose

Okay so DPBSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite band matrix and X and B are N-by-NRHS matrices. 💖

The Cholesky decomposition is used to factor A as

A = U**T * U,  if UPLO = 'U', or
A = L * L**T,  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. 🫶

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 DOUBLE PRECISION array, dimension (LDAB,N). On entry, the upper or lower triangle of the symmetric 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**T*U or A = L*L**T 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 DOUBLE PRECISION 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.

🚨 If INFO > 0, the factorization could not be completed and the solution has not been computed. That caveat is not a vibe, it's for real.

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. 💀

Definition at line 163 of file dpbsv.f.

Author

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xoxo, bestie 💖 — LAPACK Version 3.12.0, SRC/dpbsv.f(3) 🎀