SRC/sgbsv.f(3)

💅 NAME 💅

okay so this is the page for SRC/sgbsv.f, from the Library Functions Manual — and it's literally iconic, bestie. no cap.

✨ SYNOPSIS ✨

🫶 Functions/Subroutines 🫶

• subroutine sgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info) — SGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver). it's giving solver main character energy, fr fr.

🦋 Function/Subroutine Documentation 🦋

the whole vibe is this signature, periodt:

subroutine sgbsv (integer n, integer kl, integer ku, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, integer info)

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

✨ Purpose ✨

okay so SGBSV 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. lowkey genius behavior.

🌸 Parameters 🌸

• N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
• KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
• KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 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 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. it's giving storage scheme realness.
• LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
• IPIV — IPIV is 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 — 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, 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. 💀 so watch out for that.

👑 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 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. iykyk.

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

👑 Author 👑

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💖 okay bestie, that's the whole page. it's from LAPACK, Version 3.12.0, and it's literally SRC/sgbsv.f(3) — mwah! 💖