SRC/spbsv.f(3)

💅 SRC/spbsv.f ✨

okay so this is the Library Functions Manual page for SRC/spbsv.f(3), bestie — all about the spbsv subroutine. It's giving linear algebra main character energy, and yes, we're doing every detail. 🎀

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🎀 NAME

SRC/spbsv.f

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🦋 SYNOPSIS

✨ Functions/Subroutines

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

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🌸 Function/Subroutine Documentation

   subroutine spbsv (character uplo, integer n, integer kd, integer nrhs, real,
       dimension( ldab, * ) ab, integer ldab, real, dimension( ldb, * ) b,
       integer ldb, integer info)

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

🎀 Purpose

ok so SPBSV 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. Slay.

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 REAL 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 REAL 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.

Definition at line 163 of file spbsv.f.

🫶 Author

Generated automatically by Doxygen for LAPACK from the source code.

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That's the whole page, bestie 💖 — LAPACK, Version 3.12.0, source SRC/spbsv.f(3). The footer doesn't give a date, so I'm not inventing one, periodt. 🫶