ZSPSVX(3)

💅 ZSPSVX(3) 💅

okay so bestie, you've stumbled into the LAPACK corner of the man pages — welcome, it's cozy here 🫶. This one's a Library Functions Manual entry for LAPACK Version 3.12.0, and the whole thing lives in a very long file path that I'm gonna show you in a sec, no cap.

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zspsvx.f

yeah. that's the whole name. it's giving "my address is a ZIP code" 😭 but that's literally the file, so we respect it 💅

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine ZSPSVX (fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx,
    rcond, ferr, berr, work, rwork, info)

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

🎀 Function/Subroutine Documentation 🎀

🌸 the full signature, in all its glory 🌸

subroutine ZSPSVX (character fact, character uplo, integer n, integer nrhs,
    complex*16, dimension( * ) ap, complex*16, dimension( * ) afp, integer,
    dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb,
    complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond,
    double precision, dimension( * ) ferr, double precision, dimension( * )
    berr, complex*16, dimension( * ) work, double precision, dimension( * )
    rwork, integer info)

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

💖 Purpose 💖

ok so here's the vibe: ZSPSVX uses the diagonal pivoting factorization

A = U*D*U**T

A = L*D*L**T

to compute 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 🧚

Error bounds on the solution and a condition estimate are also provided. yes, it does the whole package, it's an overachiever 💅

🔥 Description 🔥

The following steps are performed, in this exact order, no skipping 👑

If FACT = 'N', 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 and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks ✨

If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below ⚠️ (yes it warns you AND keeps going, main character energy fr)
The system of equations is solved for X using the factored form of A 🎀
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it 🌈

🍓 Parameters 🍓

`FACT`

FACT is CHARACTER*1 — specifies whether or not the factored form of A has been supplied on entry.

= 'F': On entry, AFP and IPIV contain the factored form of A. AP, AFP and IPIV will not be modified. (they're just chillin, untouched 💅)
= 'N': The matrix A will be copied to AFP and factored.

`UPLO`

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

`N`

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

`NRHS`

NRHS is INTEGER — the number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

`AP`

AP is COMPLEX*16 array, dimension (N*(N+1)/2) — 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

See below for further details 👀 (there's literally a whole example down there, it's cute)

`AFP`

AFP is COMPLEX*16 array, dimension (N*(N+1)/2)

If FACT = 'F', then AFP is an input argument and on entry contains 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.
If FACT = 'N', then AFP is an output argument and on exit contains 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.

`IPIV`

IPIV is INTEGER array, dimension (N)

If FACT = 'F', then IPIV is an input argument and on entry contains 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.
If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by ZSPTRF.

`B`

B is COMPLEX*16 array, dimension (LDB,NRHS) — the N-by-NRHS right hand side matrix B.

`LDB`

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

`X`

X is COMPLEX*16 array, dimension (LDX,NRHS) — if INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X 🎉

`LDX`

LDX is INTEGER — the leading dimension of the array X. LDX >= max(1,N).

`RCOND`

RCOND is DOUBLE PRECISION — the estimate of the reciprocal condition number of the matrix A. If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0 ⚠️

`FERR`

FERR is DOUBLE PRECISION array, dimension (NRHS) — the estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error (it's a little dramatic, but honest 💅).

`BERR`

BERR is DOUBLE PRECISION array, dimension (NRHS) — the componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).

`WORK`

WORK is COMPLEX*16 array, dimension (2*N)

`RWORK`

RWORK is DOUBLE PRECISION array, dimension (N)

`INFO`

INFO is INTEGER

= 0: successful exit 🎀
< 0: if INFO = -i, the i-th argument had an illegal value 🙈
> 0: if INFO = i, and i is
- <= N: D(i,i) is exactly zero. The factorization has been completed but the factor D is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. 💀
- = N+1: D is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. ✨

👑 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 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? all squished into one linear array, it's giving "packing a suitcase with a spreadsheet" 🧳✨

⭐ Definition ⭐

Definition at line 275 of file zspsvx.f.

🫶 Author (the meta one) 🫶

Generated automatically by Doxygen for LAPACK from the source code. (not a human, just a very diligent little robot 🤖💖)

aaaand that's the whole thing bestie 🌸 so like, go forth and factor your symmetric packed matrices, you absolute icon 👑 — this page was served up by LAPACK, Version 3.12.0, straight from /home/abui...RC/zspsvx.f(3) in the footer, and honestly? slay. periodt. 💅🎀

📦 where this page comes from

opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
opensuse-leap 16.0: lapack-man 3.12.0-160000.2.2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms