ZPBSTF(3)

✨ NAME ✨

Bestie, this is ZPBSTF(3) from the Library Functions Manual — the page for the ZPBSTF subroutine, living its best life at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zpbstf.f. 💖

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine ZPBSTF (uplo, n, kd, ab, ldab, info)
    ZPBSTF

🎀 Function/Subroutine Documentation 🎀

subroutine ZPBSTF (character uplo, integer n, integer kd, complex*16,
    dimension( ldab, * ) ab, integer ldab, integer info)
    ZPBSTF

Purpose 💡

Okay so ZPBSTF computes a split Cholesky factorization of a complex Hermitian positive definite band matrix A. Slay. ✨

This routine is designed to be used in conjunction with ZHBGST.

The factorization has the form A = S**H*S where S is a band matrix of the same bandwidth as A and has this structure:

S = ( U    )
    ( M  L )

where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.

Parameters 🫶

• UPLO — CHARACTER*1
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — INTEGER
  • The order of the matrix A. N >= 0.
• KD — INTEGER
  • The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
• AB — COMPLEX*16 array, dimension (LDAB,N)
  • On entry, the upper or lower triangle of the Hermitian 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).
  • On exit, if INFO = 0, the factor S from the split Cholesky factorization A = S**H*S. See Further Details.
• LDAB — INTEGER
  • The leading dimension of the array AB. LDAB >= KD+1.
• INFO — INTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, the factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.

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 = 7, KD = 2:

S = ( s11  s12  s13                     )
    (      s22  s23  s24                )
    (           s33  s34                )
    (                s44                )
    (           s53  s54  s55           )
    (                s64  s65  s66      )
    (                     s75  s76  s77 )

If UPLO = 'U', the array AB holds:

on entry:                          on exit:

 *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53**H s64**H s75**H
 *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54**H s65**H s76**H
a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55    s66    s77

If UPLO = 'L', the array AB holds:

on entry:                          on exit:

a11  a22  a33  a44  a55  a66  a77  s11    s22    s33    s44  s55  s66  s77
a21  a32  a43  a54  a65  a76   *   s12**H s23**H s34**H s54  s65  s76   *
a31  a42  a53  a64  a64   *    *   s13**H s24**H s53    s64  s75   *    *

Array elements marked * are not used by the routine; s12**H denotes conjg(s12); the diagonal elements of S are real. 🦋

Definition at line 152 of file zpbstf.f.

Author

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