SRC/zhpsv.f(3)

✨ NAME ✨

SRC/zhpsv.f — okay bestie, that’s literally the whole NAME section, and this page is from the LAPACK Library Functions Manual. It’s giving “tiny filename, huge linear algebra energy” 💅

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🎀 SYNOPSIS 🎀

🧩 Functions/Subroutines

• subroutine zhpsv (uplo, n, nrhs, ap, ipiv, b, ldb, info) — ZHPSV computes the solution to system of linear equations A * X = B for OTHER matrices

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📖 Function/Subroutine Documentation 📖

subroutine zhpsv (character uplo, integer n, integer nrhs, complex*16, dimension( * ) ap, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZHPSV computes the solution to system of linear equations A * X = B for OTHER matrices

🎯 Purpose

okay so bestie, here’s the main-character job: ZHPSV computes the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N Hermitian matrix stored in packed format, and X and B are N-by-NRHS matrices. 🫶

The diagonal pivoting method is used to factor A as

A = U * D * U**H,  if UPLO = 'U', or
A = L * D * L**H,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then used to solve the system of equations A * X = B. ✨

🧮 Parameters

• UPLO — is CHARACTER*1
  • = 'U': upper triangle of A is stored;
  • = 'L': lower triangle of A is stored.
• N — is INTEGER
  • the number of linear equations, i.e., the order of the matrix A. N >= 0.
• NRHS — is INTEGER
  • the number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
• AP — is COMPLEX*16 array, dimension (N*(N+1)/2)
  • On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:
    • if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    • if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
  • See below for further details.
  • On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**H or A = L*D*L**H as computed by ZHPTRF, stored as a packed triangular matrix in the same storage format as A.
• IPIV — is INTEGER array, dimension (N)
  • Details of the interchanges and the block structure of D, as determined by ZHPTRF.
  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1 diagonal block.
  • If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
  • If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
• B — is COMPLEX*16 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 — is INTEGER
  • the leading dimension of the array B. LDB >= max(1,N).
• INFO — is INTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution could not be computed.

✍️ Author

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

🔍 Further Details

The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = conjg(aji))
                 a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 161 of file zhpsv.f. 📌

Author

Generated automatically by Doxygen for LAPACK from the source code.

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💖 and that’s the footer, bestie: LAPACK, Version 3.12.0, SRC/zhpsv.f(3) — no cap, all facts intact, periodt. 🎀✨