SRC/dgbsv.f(3)

Hey bestie 💖 welcome to SRC/dgbsv.f(3) — that's the title, and this page is serving from the Library Functions Manual. It's all about the dgbsv subroutine, and ngl it's a whole vibe: band matrices, LU factorization, pivots, the works. Let's get into it. ✨

🌸 NAME 🌸

SRC/dgbsv.f — literally the page name, no notes. 🎀

✨ SYNOPSIS ✨

💅 Functions/Subroutines 💅

subroutine dgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
     DGBSV computes the solution to system of linear equations A * X = B
    for GB matrices (simple driver)

💖 Function/Subroutine Documentation 💖

🦋 subroutine dgbsv — DGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver) 🦋

subroutine dgbsv (integer n, integer kl, integer ku, integer nrhs, double
    precision, dimension( ldab, * ) ab, integer ldab, integer, dimension( *
    ) ipiv, double precision, dimension( ldb, * ) b, integer ldb, integer
    info)

DGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver).

🎯 Purpose: 🎯

Okay so picture this: DGBSV computes the solution to a real system of linear equations A * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as A = L * U, where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals. The factored form of A is then used to solve the system of equations A * X = B. 🫶

🧮 Parameters 🧮

Here's the whole parameter lore, bestie:

• N — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
• KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
• KU — INTEGER. The number of superdiagonals within the band of A. KU >= 0.
• NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• AB — DOUBLE PRECISION array, dimension (LDAB,N).
  • On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL).
  • On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details.
• LDAB — INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
• IPIV — INTEGER array, dimension (N). The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i).
• B — 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
• INFO — INTEGER.
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed.

⚠️ If INFO > 0, the factorization finished but U is exactly singular, so the solution was not computed. That's a real "bestie, no" moment. 😭

🫶 Author 🫶

These fine institutions made it happen:

• 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 M = N = 6, KL = 2, KU = 1:

On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   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
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges. ✨

📍 Definition at line 161 of file dgbsv.f. 📍

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

Okay that's the whole page, bestie — it's LAPACK Version 3.12.0, SRC/dgbsv.f(3), and it ate, no crumbs. Bye! 💅✨🫶