SRC/cppsvx.f(3)

✨ NAME ✨

okay so this is the SRC/cppsvx.f page from the Library Functions Manual — section (3), bestie 💖. We're talking about the cppsvx subroutine today, and she's here to solve linear systems with packed Hermitian positive-definite matrices.

🌸 SYNOPSIS 🌸

🧚 Functions/Subroutines

       subroutine cppsvx (fact, uplo, n, nrhs, ap, afp, equed, s, b, ldb, x,
           ldx, rcond, ferr, berr, work, rwork, info)
            CPPSVX computes the solution to system of linear equations A * X =
           B for OTHER matrices

The headline is literally CPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices. Yes, OTHER is doing its own thing and we're keeping it exactly as written 😌.

💻 FUNCTION/SUBROUTINE DOCUMENTATION 💻

       subroutine cppsvx (character fact, character uplo, integer n, integer nrhs,
           complex, dimension( * ) ap, complex, dimension( * ) afp, character
           equed, real, dimension( * ) s, complex, dimension( ldb, * ) b, integer
           ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real,
           dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * )
           work, real, dimension( * ) rwork, integer info)

Same energy: CPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices.

🫶 Purpose

CPPSVX is giving Cholesky-factorization main character energy ✨. She uses either A = U**H*U or A = L*L**H to solve the complex system of linear equations

A * X = B,

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

She also hands you error bounds on the solution plus a condition estimate. Obsessed.

🧚 Description

The full glow-up sequence is:

1. If FACT = 'E', real scaling factors are computed to equilibrate the system:

diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B

Whether the system gets equilibrated depends on how A is scaled, but if equilibration happens, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the Cholesky decomposition factors A (after equilibration if FACT = 'E') as:

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

where U is an upper triangular matrix, L is a lower triangular matrix, and **H means conjugate transpose. Periodt.

3. If the leading principal minor of order i is not positive, the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of 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. She's dramatic but functional 💅.
4. The system of equations is solved for X using the factored form of A.
5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
6. If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration.

🎀 Parameters

FACT

FACT is CHARACTER*1

Tells us whether the factored form of matrix A is supplied on entry, and if not, whether A should be equilibrated before it is factored. The options are:

• = 'F' — On entry, AFP contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AP and AFP will not be modified.
• = 'N' — The matrix A will be copied to AFP and factored.
• = 'E' — The matrix A will be equilibrated if necessary, then 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 array, dimension (N*(N+1)/2)

On entry, this is the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. Exception: if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). The j-th column of A is stored in 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. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).

AFP

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

• If FACT = 'F', then AFP is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H, in the same storage format as A. If EQUED .ne. 'N', then AFP is the factored form of the equilibrated matrix A.
• If FACT = 'N', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H * U or A = L * L**H of the original matrix A.
• If FACT = 'E', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the equilibrated matrix A (see the description of AP for the form of the equilibrated matrix).

EQUED

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

• = 'N' — No equilibration (always true if FACT = 'N').
• = 'Y' — Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S).

EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.

S

S is REAL array, dimension (N)

The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive.

B

B is COMPLEX array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B.

LDB

LDB is INTEGER

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

X

X is COMPLEX array, dimension (LDX,NRHS)

If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.

LDX

LDX is INTEGER

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

RCOND

RCOND is REAL

The estimate of the reciprocal condition number of the matrix A after equilibration (if done). 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 REAL 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.

BERR

BERR is REAL 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 array, dimension (2*N)

That's the whole vibe: a COMPLEX workspace array with dimension (2*N).

RWORK

RWORK is REAL array, dimension (N)

A REAL workspace array with dimension (N).

INFO

INFO is INTEGER

The exit-status bestie:

• = 0 — successful exit.
• < 0 — if INFO = -i, the i-th argument had an illegal value.
• > 0 — if INFO = i, and i is:
  • <= N — the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned.
  • = N+1 — U 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 🌈

Ok so like, picture it: the packed storage scheme is genuinely so cute once you see it laid out. Take this example where N = 4 and UPLO = 'U' 💅

First, the two-dimensional storage of the Hermitian matrix A — look at all that wasted space in the lower triangle, we literally don't need it bestie:

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

See that note? aij = conjg(aji) — that's the whole Hermitian thing, the matrix is its own conjugate transpose, so the bottom half is just the top half's reflection and we said "not today" to storing it twice ✨

So instead, we pack the upper triangle of A all snug into one precious little array:

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

Column by column, upper triangle only, zero wasted slots. It's giving "closet organized by a Virgo" 🎀 Iconic behavior, honestly.

Ngl the packed storage scheme is the main character here — every one of those ten values is doing its job and not one of them is filler 💖

And that's a wrap on this one — Definition at line 309 of file cppsvx.f.

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👑 AUTHOR 👑

Generated automatically by Doxygen for LAPACK from the source code. No human author to thank, it's just the doc-generator doing its thing, slay 🫶

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That's the whole page bestie!! 💜 This little beauty came to you courtesy of LAPACK, Version 3.12.0, and the file SRC/cppsvx.f(3) — that's the page footer, and yes, I'm mentioning it because it deserves its moment 🌸 Byeee, go read some more manpages, ily!! 🦋