SRC/sgbtrf.f(3)

okay bestie, welcome to the LAPACK Library Functions Manual page for SRC/sgbtrf.f(3) — yes, this whole page is about one Fortran subroutine and it's lowkey iconic. 💅 No cap, we're about to go through every single parameter, author, and example, so grab a strawberry drink 🍓 and let's get into it.

💖 NAME 💖

SRC/sgbtrf.f — that's the file name on the page, bestie. The actual subroutine being documented is sgbtrf, and its docs are literally right below. 🫶

🌈 SYNOPSIS 🌈

🧚 Functions/Subroutines 🧚

• subroutine sgbtrf (m, n, kl, ku, ab, ldab, ipiv, info) — SGBTRF

💅 Function/Subroutine Documentation 💅

✨ subroutine sgbtrf ✨

here's the full signature, copied exactly because we do not mess with code, periodt:

subroutine sgbtrf (integer m, integer n, integer kl, integer ku, real, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, integer info)

SGBTRF — that's the label on the tin. 🎀

🎯 Purpose 🎯

SGBTRF computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges. ✨

This is the blocked version of the algorithm, calling Level 3 BLAS. It's giving blocked-algorithm main character energy, ngl. 🔥

🧮 Parameters 🧮

okay so here's the parameter vibe check, one by one, with all the constraints intact:

• M — M is INTEGER. The number of rows of the matrix A. M >= 0.
• N — N is INTEGER. The number of columns 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.
• 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(m,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 — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
• IPIV — IPIV is INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
• 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 division by zero will occur if it is used to solve a system of equations.

👑 Author 👑

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

🔥 Further Details 🔥

okay so 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. That's the whole tea. 🍵

Definition at line 143 of file sgbtrf.f.

👑 Author 👑

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