CPPSVX(3)

CPPSVX(3) — Library Functions Manual, but make it cute 💅

Hey bestie, welcome to the Library Functions Manual page for CPPSVX(3). The official NAME line is literally a file path, so we're starting with that vibe. 🫶

✨ NAME ✨

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

That's the whole NAME section. Iconic, honestly. 💀

🌸 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

💻 Function/Subroutine Documentation 💻

🧚 subroutine CPPSVX

   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)

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

🎯 Purpose

Okay so CPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a 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. 💖

It also hands you error bounds on the solution and a condition estimate. Slay. ✨

📝 Description

The following steps are performed:

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 or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix 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 indicates conjugate transpose.

3. If the leading principal minor of order i is not positive, 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.
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. It specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.

• = '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, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array, except 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 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. 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. It 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).

RWORK — RWORK is REAL 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: 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

(No further details are listed in this section on the page.)

...and this is the part where the packed storage scheme gets a whole little diagram moment, bestie — it's giving "I made a visual aid in the group chat" energy, and honestly? Iconic. 💅

The packed storage scheme, illustrated — the docs literally say so, and they're right, because this is the example where N = 4 and UPLO = 'U':

Two-dimensional storage of the Hermitian matrix A:

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

See how the upper triangle is all that's actually there? The lower part is just the conjugate mirror — we don't need to store it twice, that's so extra. 🙈

Packed storage of the upper triangle of A:

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

Column by column, one triangle only, no waste. We love an efficient queen. 👑 Note the (aij = conjg(aji)) up there in the matrix — that's the Hermitian condition, fr fr: every element below the diagonal is the complex conjugate of its mirror above it. 💖

Definition at line 309 of file cppsvx.f.

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🌸 Author 🌸

Generated automatically by Doxygen for LAPACK from the source code. Yup — nobody hand-wrote this one, it's Doxygen doing its little robot thing and honestly it ate. 🎀

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Sign-off time, bestie 🫶 This was LAPACK, Version 3.12.0, and the page you just read lives at /home/abui...RC/cppsvx.f(3). Go forth and factor your Hermitian matrices in packed storage, my main character. Periodt. ✨💜🦋