ZHPSVX(3)

💖 ZHPSVX(3) 💖

okay bestie, welcome to the zhpsvx.f page — it lives in the Library Functions Manual, and the file itself is chilling at:

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

that's the whole NAME situation, yes it's literally just a path, slay 🎀

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✨ SYNOPSIS ✨

Functions/Subroutines ✨

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

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

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🌸 Function/Subroutine Documentation 🌸

subroutine ZHPSVX (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)

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

💜 Purpose 💜

ZHPSVX uses the diagonal pivoting factorization A = U*D*U**H or A = L*D*L**H to compute 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. Error bounds on the solution and a condition estimate are also provided.

like, it literally solves your system AND gives you receipts on how wrong it might be 😭 iconic

🌈 Description 🌈

The following steps are performed: 1. If FACT = 'N', 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. 2. 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. 3. The system of equations is solved for X using the factored form of A. 4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

a whole four-step glow-up routine, we love a process 👑

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🎀 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. AFP and IPIV will not be modified. - = '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 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. See below for further details. 👀

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**H or A = L*D*L**H as computed by ZHPTRF, 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**H or A = L*D*L**H as computed by ZHPTRF, stored as a packed triangular matrix in the same storage format as A.

in-out energy depending on FACT, main character behavior honestly 🫶

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 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. 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 ZHPTRF.

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.

ngl "almost always a slight overestimate" is so real of it 🙈

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.

⚠️ bestie watch that INFO = N+1 case — it's a warning, not a vibe. The matrix is singular to working precision but the routine still hands you the solution and error bounds, so don't just assume it's toast.

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🦋 Author 🦋

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

a whole collab, the group project actually got finished 😭💖

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⭐ 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 ]

see how it just stacks the columns into one long array?? lowkey satisfying ✨ also note aij = conjg(aji) — Hermitian, so the mirror image is the conjugate, not a straight copy, iykyk 💅

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Definition at line 275 of file zhpsvx.f.

🌸 Author 🌸

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the page, bestie 🎀 generated by Doxygen for LAPACK, Version 3.12.0 — source file zhpsvx.f, and yes, it's the same path we started with because the universe is a circle 🌈💜