SRC/cspsv.f(3)

💅 SRC/cspsv.f — like, the LAPACK manual page, but make it make sense

okay so this is cspsv.f, straight out of the Library Functions Manual, and bestie… it's a Fortran subroutine page. don't scroll away yet, i promise it's actually iconic once you know what it does 🫶

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✨ NAME ✨

SRC/cspsv.f — that's the whole name section, no cap, the manpage really just says that 💀

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🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine cspsv (uplo, n, nrhs, ap, ipiv, b, ldb, info)

CSPSV computes the solution to system of linear equations A * X = B for OTHER matrices — yes that's the literal tagline, "OTHER matrices," we love a mysterious queen 👑

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🦋 Function/Subroutine Documentation 🦋

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

so, what's the vibe? CSPSV computes the solution to system of linear equations A * X = B for OTHER matrices ✨

🎀 Purpose 🎀

!>
!> CSPSV computes the solution to a complex system of linear equations
!>    A * X = B,
!> where A is an N-by-N symmetric 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**T,  if UPLO = 'U', or
!>    A = L * D * L**T,  if UPLO = 'L',
!> where U (or L) is a product of permutation and unit upper (lower)
!> triangular matrices, D is symmetric 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.
!>

so basically: A is N-by-N, symmetric, and stored packed (more on that chaos below 🫠), X and B are N-by-NRHS. the diagonal pivoting method factors A into A = U * D * U**T when UPLO = 'U', or A = L * D * L**T when UPLO = 'L'. U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. then the factored form of A gets used to solve A * X = B. slay, that's the whole pipeline 💅

🌈 Parameters 🌈

• UPLO — CHARACTER*1. which triangle of A are we storing, bestie?
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — INTEGER. the number of linear equations, i.e., the order of the matrix A. constraint: N >= 0. ✨
• NRHS — INTEGER. the number of right hand sides, i.e., the number of columns of the matrix B. constraint: NRHS >= 0.
• AP — COMPLEX array, dimension (N*(N+1)/2).
  • On entry: the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. the j-th column of A is stored in AP like this:
    • 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**T or A = L*D*L**T as computed by CSPTRF, stored as a packed triangular matrix in the same storage format as A.
• IPIV — INTEGER array, dimension (N). details of the interchanges and the block structure of D, as determined by CSPTRF. so here's the tea ☕:
  • 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 — COMPLEX 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 — INTEGER. the leading dimension of the array B. constraint: LDB >= max(1,N). no shortcuts here, she means it.
• INFO — INTEGER. the receipts:
  • = 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. so close and yet so far 💀

fr fr, watch that INFO > 0 case. singular D means no solution for you, bestie.

👑 Author 👑

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

the whole squad showed up 🫶

🧚 Further Details: 🧚

!>
!>  The packed storage scheme is illustrated by the following example
!>  when N = 4, UPLO = 'U':
!>
!>  Two-dimensional storage of the symmetric matrix A:
!>
!>     a11 a12 a13 a14
!>         a22 a23 a24
!>             a33 a34     (aij = aji)
!>                 a44
!>
!>  Packed storage of the upper triangle of A:
!>
!>  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
!>

see how it's giving "squished but still symmetric"? (aij = aji), so we only keep one half. lowkey elegant 🌸

Definition at line 161 of file cspsv.f.

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⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code. (yes, a robot wrote this, we're just vibing on top of it 🤖✨)

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and that's the whole page, bestie! 💖 this was SRC/cspsv.f from LAPACK Version 3.12.0 — go forth and solve your symmetric packed complex systems, main character style 👑🌀