SRC/zgbtrf.f(3)

💖 NAME

okay bestie, welcome to SRC/zgbtrf.f from the Library Functions Manual — this is the ZGBTRF page and it's lowkey iconic linear-algebra behavior. 💅

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✨ SYNOPSIS

🫶 Functions/Subroutines

   Functions/Subroutines
       subroutine zgbtrf (m, n, kl, ku, ab, ldab, ipiv, info)
           ZGBTRF

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🎀 Function/Subroutine Documentation

Here's the actual callable, bestie:

   subroutine zgbtrf (integer m, integer n, integer kl, integer ku, complex*16,
       dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv,
       integer info)
       ZGBTRF

🌸 Purpose

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

This is the blocked version of the algorithm, calling Level 3 BLAS. no cap.

🦋 Parameters

💜 M

M is INTEGER. It's the number of rows of the matrix A. M >= 0.

💜 N

N is INTEGER. It's the number of columns of the matrix A. N >= 0.

💜 KL

KL is INTEGER. It's the number of subdiagonals within the band of A. KL >= 0.

💜 KU

KU is INTEGER. It's the number of superdiagonals within the band of A. KU >= 0.

💜 AB

AB is COMPLEX*16 array, dimension (LDAB,N).

On entry, this is 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, it holds 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. It's the leading dimension of the array AB. LDAB >= 2*KL+KU+1.

💜 IPIV

IPIV is INTEGER array, dimension (min(M,N)).

These are 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. literally do not ignore this one, bestie 😭

🦋 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. 💀✨

Definition at line 143 of file zgbtrf.f.

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👑 Author

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bye bestie 💖 — LAPACK Version 3.12.0, page SRC/zgbtrf.f(3).