SRC/zspsv.f(3)

💅 SRC/zspsv.f — from the Library Functions Manual ✨

okay bestie, sit down, because this one is a whole moment. This is SRC/zspsv.f, straight out of the Library Functions Manual, and it's about to solve your linear system AND your skin routine. 🧖‍♀️

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines

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

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🫶 Function/Subroutine Documentation

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

ZSPSV computes the solution to system of linear equations A * X = B for OTHER matrices. slay. 💅

🌸 Purpose:

!>
!> ZSPSV 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: factor A with the diagonal pivoting method, then use that factored form to solve A * X = B. it's giving organized. 💖

👑 Parameters

UPLO

!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored.
!>

pick your fighter: 'U' for the upper triangle, 'L' for the lower one. no in-between, bestie. 🦋

N

!>          N is INTEGER
!>          The number of linear equations, i.e., the order of the
!>          matrix A.  N >= 0.
!>

the number of linear equations — i.e. the order of the matrix A. it must be N >= 0, so no negative chaos allowed. ✨

NRHS

!>          NRHS is INTEGER
!>          The number of right hand sides, i.e., the number of columns
!>          of the matrix B.  NRHS >= 0.
!>

how many right hand sides you got, aka how many columns B has. NRHS >= 0. 🎀

AP

!>          AP is COMPLEX*16 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 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**T or A = L*D*L**T as computed by ZSPTRF, stored as
!>          a packed triangular matrix in the same storage format as A.
!>

COMPLEX*16 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, bestie, there's a whole diagram 💀)

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 ZSPTRF, stored as a packed triangular matrix in the same storage format as A. it literally stays in the same format, iconic behavior. 👑

IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          Details of the interchanges and the block structure of D, as
!>          determined by ZSPTRF.  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.
!>

INTEGER array, dimension (N). the details of the interchanges and the block structure of D, as determined by ZSPTRF. read carefully, this is the lore:

• 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.

yes the negative signs matter. no you cannot skip them. 💅

B

!>          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.
!>

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. glow up! ✨

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,N).
!>

INTEGER. the leading dimension of the array B, and it must satisfy LDB >= max(1,N). 🫶

INFO

!>          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.
!>

INTEGER, and it's your vibe check on the whole operation:

• = 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. 💀 it did all that work and then couldn't finish. heartbreaking.

📜 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

(academic royalty, all of them 👑)

🦋 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 ]
!>

ok so here's the visual aid, because reading about packed storage without a picture is lowkey painful. 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 just squishes into one line? that's the whole trick. ate and left no crumbs. 🍓

Definition at line 161 of file zspsv.f.

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💖 Author

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole page, bestie 🫶 this was LAPACK, Version 3.12.0, live from SRC/zspsv.f(3) — go forth and solve your A * X = B, you've got this 🌈💜