hpsvx(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

okay bestie, welcome to the hpsvx(3) glow-up ✨ — this is the Library Functions Manual page for the <rainbow>{hp,sp}svx</rainbow> crew, and the vibe is literally “factor and solve, expert.” 💅

✨ NAME ✨

hpsvx — {hp,sp}svx: factor and solve, expert. 🎀

🌸 SYNOPSIS 🌸

💖 Functions 💖

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

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

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

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

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

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

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

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

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

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

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

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

🔍 Detailed Description 🔍

This section is literally just a heading in the source, bestie — the actual deets are living their best life in Function Documentation right below. 🫶

🧮 Function Documentation 🧮

💖 subroutine `chpsvx` 💖

subroutine chpsvx (character fact, character uplo, integer n, integer nrhs,
    complex, dimension( * ) ap, complex, dimension( * ) afp, integer,
    dimension( * ) ipiv, 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)

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

✨ Purpose ✨

CHPSVX 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.

🎀 Description 🎀

The following steps are performed:

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.
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.
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. 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 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 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 CHPTRF, 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 CHPTRF, 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 CHPTRF.

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 CHPTRF.

B — B is COMPLEX 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 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 REAL. 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 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: 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, if RCOND is tiny or zero, that matrix is singular to working precision — read the INFO tea before trusting the solve. 🚨💅

👑 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 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 ]

Definition at line 275 of file chpsvx.f.

subroutine cspsvx

subroutine cspsvx (character fact, character uplo, integer n, integer nrhs,
    complex, dimension( * ) ap, complex, dimension( * ) afp, integer,
    dimension( * ) ipiv, 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)

okay so bestie, CSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices 💅 (and yes the all-caps "OTHER" is doing main character energy, we respect it).

Purpose 💖

!>
!> CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
!> 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.
!>

Translation: CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to solve 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. It also hands you error bounds on the solution plus a condition estimate. So generous, honestly 🫶.

Description 🌸

The following steps are performed:

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. (she literally warns you and then does the thing anyway, iconic)
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 — 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.
- = 'N': The matrix A will be copied to AFP and factored.
UPLO — is CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS — is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
AP — is COMPLEX 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. 👀

AFP — is COMPLEX 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 CSPTRF, 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 CSPTRF, stored as a packed triangular matrix in the same storage format as A. (same storage format as A, she said, twice, because it matters 💀)
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 CSPTRF. 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 CSPTRF.
B — is COMPLEX array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B.
LDB — is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
X — is COMPLEX array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. (the payoff, the moment, the slay)
LDX — is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
RCOND — is REAL. 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 — 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. (she's a lil dramatic, she overestimates, we love her for it)
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 — is COMPLEX array, dimension (2*N).
RWORK — is REAL array, dimension (N).
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. (she computes it anyway bc sometimes the vibe check is wrong, no cap)

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 how she just... folds it up? diagonal by diagonal? obsessed 💖

Definition at line 275 of file cspsvx.f.

subroutine dspsvx

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

DSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices ✨ — yes it's the real-valued sibling of the one above, same energy, different precision.

Purpose 💜

!>
!> DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
!> A = L*D*L**T to compute the solution to a real 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.
!>

so DSPSVX pulls the same diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to solve a real system A * X = B, where A is an N-by-N symmetric matrix in packed format and X and B are N-by-NRHS matrices. Error bounds and a condition estimate on the solution? Also provided. She came prepared 🎀.

Description 🌈

(hold that thought — the steps are literally right there, next part, don't come for me 😭)

okay so the step-by-step for DSPSVX is giving main character energy:

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.
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.
- = '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 DOUBLE PRECISION 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.

AFP — AFP is DOUBLE PRECISION 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 DSPTRF, 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 DSPTRF, 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 DSPTRF.
- 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 DSPTRF.
B — B is DOUBLE PRECISION 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 DOUBLE PRECISION 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.
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 DOUBLE PRECISION array, dimension (3*N).
IWORK — IWORK is INTEGER 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 ]

Definition at line 274 of file dspsvx.f.

🧚 subroutine sspsvx 🧚

   subroutine sspsvx (character fact, character uplo, integer n, integer nrhs,
       real, dimension( * ) ap, real, dimension( * ) afp, integer, dimension( *
       ) ipiv, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx,
       * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real,
       dimension( * ) berr, real, dimension( * ) work, integer, dimension( * )
       iwork, integer info)

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

✨ Purpose ✨

SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real 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.

🎀 Description 🎀

The following steps are performed:

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.
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.
- = 'N': The matrix A will be copied to AFP and factored.
UPLO
UPLO — ok so UPLO is a CHARACTER*1, and it's literally just picking a triangle. 'U' means the upper triangle of A is stored; 'L' means the lower triangle of A is stored. That's the whole vibe 💅
N — N is INTEGER, bestie. It's the number of linear equations, i.e., the order of the matrix A. And the rule is N >= 0 — no negative nonsense 🫶
NRHS — NRHS is INTEGER too. It's the number of right hand sides, i.e., the number of columns of the matrices B and X. Requirement: NRHS >= 0 ✨
AP — AP is a REAL array, dimension (N*(N+1)/2). It's 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 👀
AFP — AFP is a REAL 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 SSPTRF, 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 SSPTRF, stored as a packed triangular matrix in the same storage format as A. Same energy, different direction 😌

IPIV — IPIV is an 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 SSPTRF.
- 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 SSPTRF. Basically it's the receipts of what got swapped 🔁

B — B is a REAL array, dimension (LDB,NRHS). It's the N-by-NRHS right hand side matrix B. The main character of the equation, lowkey 💖
LDB — LDB is INTEGER. It's the leading dimension of the array B, and you need LDB >= max(1,N). Respect the bounds, bestie 🫡
X — X is a REAL array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X. The moment we've all been waiting for ✨
LDX — LDX is INTEGER. It's the leading dimension of the array X, and the rule is LDX >= max(1,N). Keep it valid or it's giving illegal argument energy 😭
RCOND — RCOND is REAL. It's 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.

So if RCOND is tiny or zero, that's your red flag era. The routine is literally telling you the matrix is singular to working precision, and INFO > 0 is the clue 💀

FERR — FERR is a REAL array, dimension (NRHS). It's 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. So it's a little dramatic, but in a helpful way 💅
BERR — BERR is a REAL array, dimension (NRHS). It's 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). Tiny number, big meaning 🎀
WORK — WORK is a REAL array, dimension (3*N). It's the workspace, doing background support like a true bestie 🫶
IWORK — IWORK is an INTEGER array, dimension (N). More workspace, but integer flavored ✨
INFO — INFO is INTEGER, and this is the status report:
- = 0: successful exit. Slay.
- < 0: if INFO = -i, the i-th argument had an illegal value. Not iconic.
- > 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.

The INFO = N+1 case is giving "warning, but we still pulled through" energy. The routine still computes the solution and error bounds because sometimes the computed solution is more accurate than 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 ]

Definition at line 274 of file sspsvx.f.

🦋 subroutine `zhpsvx` 🦋

   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. Yes, it's thorough, and we love that for us 💖

🌈 Description 🌈

The following steps are performed:

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.

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.
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. It 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. Pick your lane, bestie 💅
UPLO — UPLO is CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored. Triangle choice, but make it Hermitian ✨
N — N is INTEGER. It's the number of linear equations, i.e., the order of the matrix A. N >= 0, always 🫶
NRHS — NRHS is INTEGER. It's the number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0, no cap 💯
AP — AP is a COMPLEX*16 array, dimension (N*(N+1)/2). It's 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 — okay so AFP is a COMPLEX*16 array, dimension (N*(N+1)/2). If FACT = 'F', then AFP is an input argument and on entry it 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 it contains that same 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. slay, storage format twins 💅
IPIV — IPIV is an INTEGER array, dimension (N). If FACT = 'F', then IPIV is an input argument and on entry it contains details of the interchanges and the block structure of D, as determined by ZHPTRF. iykyk, this is the receipts 🧾
- 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 it contains details of the interchanges and the block structure of D, as determined by ZHPTRF.
B — B is a COMPLEX*16 array, dimension (LDB,NRHS). It's the N-by-NRHS right hand side matrix B. the main character of the equation, honestly 🫶
LDB — LDB is INTEGER. It's the leading dimension of the array B, and LDB >= max(1,N). respect the leading dimension or catch hands 🫵
X — X is a COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X. the answer era ✨
LDX — LDX is INTEGER. It's the leading dimension of the array X, and LDX >= max(1,N). same energy as LDB, no cap 🍑
RCOND — RCOND is DOUBLE PRECISION. It's 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. bestie, that's the “uh oh” signal 🚨
FERR — FERR is a DOUBLE PRECISION array, dimension (NRHS). It's 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. generous king, giving us an upper bound 💜
BERR — BERR is a DOUBLE PRECISION array, dimension (NRHS). It's 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). backward error but make it fashion 💫
WORK — WORK is a COMPLEX*16 array, dimension (2*N). scratch space, doing behind-the-scenes work 🛠️
RWORK — RWORK is a DOUBLE PRECISION array, dimension (N). more scratch space, real-valued this time 🎀
INFO — INFO is INTEGER. the vibe check at the end 💅
- = 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. plot twist: sometimes the “bad” condition number still eats 😭🔥

Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

shoutout to the Unix aunties and uncles who wrote this down, fr fr 🌿

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 ]

visual learner moment: that's the whole packed-storage wig in one example 🎀

Definition at line 275 of file zhpsvx.f.

🦋 subroutine `zspsvx` 🦋

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. “OTHER matrices” is giving mysterious side quest, but okay 👀

🎀 Purpose: 🎀

ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or 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. accuracy? we love her 💖

💅 Description: 💅

The following steps are performed:

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. block diagonal D is the moment, honestly 🧩
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. warning bestie: singular D means no solution/error bounds, periodt 🚫
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. glow-up step: refine, refine, refine 💅

✨ Parameters ✨

FACT — FACT is CHARACTER*1. It 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.
- = '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. It's the number of linear equations, i.e., the order of the matrix A. N >= 0.
NRHS — NRHS is INTEGER. It's the number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
AP — AP is a COMPLEX*16 array, dimension (N*(N+1)/2). It's 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. packed columnwise, iconic storage behavior 🍓

AFP — AFP is a COMPLEX*16 array, dimension (N*(N+1)/2). If FACT = 'F', then AFP is an input argument and on entry it 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 it 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. ZSPTRF said “I'll handle the factorization,” and honestly? she did 💜
IPIV

okay bestie, picking up right where we left off with IPIV — no cap, this one's a little extra 💅

IPIV — 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 — COMPLEX*16 array, dimension (LDB,NRHS). The N-by-NRHS right hand side matrix B. 💖
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N). slay, keep it legal ✨
X — COMPLEX*16 array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. 🌸
LDX — INTEGER. The leading dimension of the array X. LDX >= max(1,N). periodt 💅
RCOND — 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. lowkey important 🚨
FERR — 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, it's giving "trust but verify" 😭
BERR — 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 — COMPLEX*16 array, dimension (2*N). 💜
RWORK — DOUBLE PRECISION array, dimension (N). 🦋
INFO — 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. not me being dramatic but this is actually iconic 🔥

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? it's literally just the upper triangle tucked into one cute lil array 🎀✨

Definition at line 275 of file zspsvx.f.

Author (again, but make it Doxygen) 💻

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole entire page bestie, no cap 🫶 — straight from LAPACK, Version 3.12.0, hpsvx(3), generated with love (and Doxygen). byeeee 💅✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

hpsvx(3)

✨ NAME ✨

🌸 SYNOPSIS 🌸

💖 Functions 💖

🔍 Detailed Description 🔍

🧮 Function Documentation 🧮

💖 subroutine chpsvx 💖

✨ Purpose ✨

🎀 Description 🎀

🦋 Parameters 🦋

👑 Author 👑

🌈 Further Details 🌈

subroutine cspsvx

Purpose 💖

Description 🌸

✨ Parameters ✨

Author 👑

Further Details: 🦋

subroutine dspsvx

Purpose 💜

Description 🌈

🌸 Parameters 🌸

💖 Author 💖

🌈 Further Details 🌈

🧚 subroutine sspsvx 🧚

✨ Purpose ✨

🎀 Description 🎀

🍓 Parameters 🍓

👑 Author 👑

🔍 Further Details: 🔍

🦋 subroutine zhpsvx 🦋

🎯 Purpose 🎯

🌈 Description 🌈

🧾 Parameters 🧾

Author 🫶

Further Details: 🌸

🦋 subroutine zspsvx 🦋

🎀 Purpose: 🎀

💅 Description: 💅

✨ Parameters ✨

Author 👑

Further Details: 🧚

Author (again, but make it Doxygen) 💻

💖 subroutine `chpsvx` 💖

🦋 subroutine `zhpsvx` 🦋

🦋 subroutine `zspsvx` 🦋