doublePOsolve(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

💖 doublePOsolve(3) — LAPACK's PO solve squad 💖

okay so bestie, you opened the manual page for doublePOsolve(3), and it's literally the group of double solve driver functions for PO matrices. No cap, this is where Cholesky pulls up and handles your symmetric positive definite system. Let's go. ✨

🌸 NAME 🌸

doublePOsolve — double 💅

✨ SYNOPSIS ✨

🎀 Functions 🎀

dposv — subroutine dposv (UPLO, N, NRHS, A, LDA, B, LDB, INFO) — DPOSV computes the solution to system of linear equations A * X = B for PO matrices 🫶
dposvx — subroutine dposvx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) — DPOSVX computes the solution to system of linear equations A * X = B for PO matrices 💖
dposvxx — subroutine dposvxx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO) — DPOSVXX computes the solution to system of linear equations A * X = B for PO matrices 🔥
dsposv — subroutine dsposv (UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, SWORK, ITER, INFO) — DSPOSV computes the solution to system of linear equations A * X = B for PO matrices 🌈

💫 Detailed Description 💫

This is the group of double solve driver functions for PO matrices. That's the whole vibe. PO matrices are the main character here, and these routines are the besties who solve them. 👑

👑 Function Documentation 👑

🧚 subroutine `dposv` 🧚

subroutine dposv (character UPLO, integer N, integer NRHS, double precision,
    dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, *
    ) B, integer LDB, integer INFO)

DPOSV computes the solution to system of linear equations A * X = B for PO matrices. ✨

🎯 Purpose

DPOSV computes the solution to a real system of linear equations

   A * X = B,

where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as

   A = U**T* U,  if UPLO = 'U', or
   A = L * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used to solve the system of equations A * X = B. 💅

🧾 Parameters

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 matrix B. NRHS >= 0.
A — A is DOUBLE PRECISION array, dimension (LDA,N)
- On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
- On exit, if INFO = 0, the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.
LDA — LDA is INTEGER
- The leading dimension of the array A. LDA >= max(1,N).
B — B is DOUBLE PRECISION array, dimension (LDB,NRHS)
- On entry, the N-by-NRHS right hand side matrix B.
- On exit, if INFO = 0, the N-by-NRHS solution matrix X.
LDB — LDB is INTEGER
- The leading dimension of the array B. LDB >= max(1,N).
INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

🫶 Author

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

🦋 subroutine `dposvx` 🦋

subroutine dposvx (character FACT, character UPLO, integer N, integer NRHS,
    double precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldaf, * ) AF, integer LDAF, character EQUED, double
    precision, dimension( * ) S, 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)

DPOSVX computes the solution to system of linear equations A * X = B for PO matrices. 💖

🎯 Purpose

DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations

   A * X = B,

where A is an N-by-N symmetric positive definite matrix 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 = '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.

If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

   A = U**T* U,  if UPLO = 'U', or
   A = L * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is a lower triangular matrix.

If the leading i-by-i principal minor is not positive definite, 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.
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
- 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, AF contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. A and AF will not be modified.
- = 'N': The matrix A will be copied to AF and factored.
- = 'E': The matrix A will be equilibrated if necessary, then copied to AF 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.
A — A is DOUBLE PRECISION array, dimension (LDA,N)
- On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. 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).
LDA — LDA is INTEGER
- The leading dimension of the array A. LDA >= max(1,N).
AF — AF is DOUBLE PRECISION array, dimension (LDAF,N)
- If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S).
- If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A.
- If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).
LDAF — LDAF is INTEGER
- The leading dimension of the array AF. LDAF >= max(1,N).
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 DOUBLE PRECISION 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`

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, she's the N-by-NRHS right hand side matrix B 💖.

On exit it depends on EQUED, bestie:

if EQUED = 'N', B is not modified — untouched, unbothered, no notes ✨
if EQUED = 'Y', B is overwritten by diag(S) * B — so yeah, she's been through something 💅

`LDB`

LDB is INTEGER

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

`X`

X is DOUBLE PRECISION array, dimension (LDX,NRHS)

If INFO = 0 or INFO = N+1, this is the N-by-NRHS solution matrix X to the original system of equations 👑.

Heads up though — if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X. So like… read the fine print, bestie 😭

`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 after equilibration (if it happened at all) 💅.

If RCOND is less than the machine precision (in particular, if RCOND = 0), then the matrix is singular to working precision. That whole situation gets flagged 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), then 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) 📏.

And ngl, the estimate is exactly as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. She's a little dramatic, that's the vibe 💜.

`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)

The scratch space. Not the main character, but absolutely carrying the plot 🫶.

`IWORK`

IWORK is INTEGER array, dimension (N)

More scratch space, but integer flavoured 🍓.

`INFO`

INFO is INTEGER

The receipts, bestie 📜:

= 0: successful exit — slay ✅
< 0: if INFO = -i, the i-th argument had an illegal value 🙈
> 0: if INFO = i, and i is…
- <= N: the leading minor of order i of A is not positive definite, 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 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 defending the math but okay, go off 🔥

👑 Author

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

`subroutine dposvxx`

subroutine dposvxx (character FACT, character UPLO, integer N, integer NRHS,
    double precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldaf, * ) AF, integer LDAF, character EQUED, double
    precision, dimension( * ) S, double precision, dimension( ldb, * ) B,
    integer LDB, double precision, dimension( ldx, * ) X, integer LDX,
    double precision RCOND, double precision RPVGRW, double precision,
    dimension( * ) BERR, integer N_ERR_BNDS, double precision, dimension(
    nrhs, * ) ERR_BNDS_NORM, double precision, dimension( nrhs, * )
    ERR_BNDS_COMP, integer NPARAMS, double precision, dimension( * ) PARAMS,
    double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,
    integer INFO)

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

Purpose:

DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a double precision system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices. 💅

If requested, both normwise and maximum componentwise error bounds are returned. DPOSVXX will return a solution with a tiny guaranteed error (O(eps) where eps is the working machine precision) unless the matrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers also are calculated and returned 👑

DPOSVXX accepts user-provided factorizations and equilibration factors; see the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous DPOSVXX call will also produce a solution with either O(eps) errors or warnings, but we cannot make that claim for general user-provided factorizations and equilibration factors if they differ from what DPOSVXX would itself produce. So don't get cute with it, bestie. 😭

Description:

The following steps are performed:

If FACT = 'E', double precision 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 💫.

If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as

A = U**T* U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',

where U is an upper triangular matrix and L is a lower triangular matrix 🌸.

If the leading i-by-i principal minor is not positive definite, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A (see argument RCOND). If the reciprocal of the condition number is less than machine precision, 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 ✨.
By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), the routine will use iterative refinement to try to get a small error and error bounds. Refinement calculates the residual to at least twice the working precision 🔥.
If equilibration was used, the matrix X is premultiplied by diag(S) so that it solves the original system before equilibration 💅.

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. If the defaults are acceptable, users can pass NPARAMS = 0 which prevents the source code from accessing the PARAMS argument. Iconic low-effort move, honestly. 🎀

Parameters

`FACT`

FACT is CHARACTER*1

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, AF contains the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by S. A and AF are not modified ✨
= 'N': The matrix A will be copied to AF and factored 💫
= 'E': The matrix A will be equilibrated if necessary, then copied to AF 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 💖

`A`

A is DOUBLE PRECISION array, dimension (LDA,N)

On entry, the symmetric matrix A — except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S) 😤.

If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced — leave her alone, she's off duty 🙈.

If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

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) 💅.

`LDA`

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N). ✨

`AF`

AF is DOUBLE PRECISION array, dimension (LDAF,N)

If FACT = 'F', then AF is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AF is the factored form of the equilibrated matrix diag(S)*A*diag(S) 🌸.

If FACT = 'N', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the original matrix A 💫.

If FACT = 'E', then AF is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix) 🎀.

`LDAF`

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N). 🦋

`EQUED`

EQUED is CHARACTER*1 — tells you what kind of equilibration got done. 💫
- = 'N': no equilibration happened at all (this is always the case if FACT = 'N').
- = 'Y': both row and column equilibration went down, meaning A has been replaced by diag(S) * A * diag(S).

Heads up bestie: EQUED is an input argument if FACT = 'F'. Otherwise it's an output argument. Know the difference or perish. 🫠

S is DOUBLE PRECISION array, dimension (N) — the row scale factors for A. 🌸

If EQUED = 'Y', then A gets multiplied on the left and the right by diag(S). S is an input argument if FACT = 'F'; otherwise S is an output argument. If FACT = 'F' and EQUED = 'Y', every single element of S must be positive. If S is output, every element of S is a power of the radix. If S is input, every element of S should be a power of the radix so you get a reliable solution and reliable error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. And rounding errors during scaling mean you're refining with a matrix that is not equivalent to the input matrix, which produces error estimates that may not be reliable. 😭 Lowkey devastating but okay.

B is DOUBLE PRECISION array, dimension (LDB,NRHS) — on entry, it's 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 is INTEGER — the leading dimension of the array B. Requirement: LDB >= max(1,N). 📏
X is DOUBLE PRECISION array, dimension (LDX,NRHS) — if INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(S))*X. 👑
LDX is INTEGER — the leading dimension of the array X. Requirement: LDX >= max(1,N).
RCOND is DOUBLE PRECISION — the reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if it was done). If this is less than the machine precision (in particular, if it is zero), the matrix is singular to working precision. Note that the error may still be small even if this number is very small and the matrix appears ill-conditioned. 💀 Not the vibe you think it is.
RPVGRW is DOUBLE PRECISION — reciprocal pivot growth. On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is what's used here. If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, the estimated condition numbers, and the error bounds could be unreliable. If factorization fails with 0<INFO<=N, then this contains the reciprocal pivot growth factor for the leading INFO columns of A. 🚩
BERR is DOUBLE PRECISION array, dimension (NRHS) — componentwise relative backward error. This is 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).
N_ERR_BNDS is INTEGER — number of error bounds to return for each right hand side and each type (normwise or componentwise). See ERR_BNDS_NORM and ERR_BNDS_COMP below. ⬇️
ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) — for each right-hand side, this array holds info about various error bounds and condition numbers corresponding to the normwise relative error, which is defined like this:

Normwise relative error in the ith solution vector:
        max_j (abs(XTRUE(j,i) - X(j,i)))
       ------------------------------
             max_j abs(X(j,i))

The array is indexed by the type of error information as described below. There currently are up to three pieces of information returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith right-hand side. 🎯

The second index in ERR_BNDS_NORM(:,err) contains the following three fields:

err = 1 'Trust/don't trust' boolean. Trust the answer if the
         reciprocal condition number is less than the threshold
         sqrt(n) * dlamch('Epsilon').

err = 2 'Guaranteed' error bound: The estimated forward error,
         almost certainly within a factor of 10 of the true error
         so long as the next entry is greater than the threshold
         sqrt(n) * dlamch('Epsilon'). This error bound should only
         be trusted if the previous boolean is true.

err = 3  Reciprocal condition number: Estimated normwise
         reciprocal condition number.  Compared with the threshold
         sqrt(n) * dlamch('Epsilon') to determine if the error
         estimate is 'guaranteed'. These reciprocal condition
         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
         appropriately scaled matrix Z.
         Let Z = S*A, where S scales each row by a power of the
         radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions. 📓

ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) — for each right-hand side, this array holds info about various error bounds and condition numbers corresponding to the componentwise relative error, defined as:

Componentwise relative error in the ith solution vector:
               abs(XTRUE(j,i) - X(j,i))
        max_j ----------------------
                    abs(X(j,i))

The array is indexed by the right-hand side i (which the componentwise relative error depends on) and the type of error information as described below. There currently are up to three pieces of information returned for each right-hand side. If componentwise accuracy is not requested (PARAMS(3) = 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following three fields:

err = 1 'Trust/don't trust' boolean. Trust the answer if the
         reciprocal condition number is less than the threshold
         sqrt(n) * dlamch('Epsilon').

err = 2 'Guaranteed' error bound: The estimated forward error,
         almost certainly within a factor of 10 of the true error
         so long as the next entry is greater than the threshold
         sqrt(n) * dlamch('Epsilon'). This error bound should only
         be trusted if the previous boolean is true.

err = 3  Reciprocal condition number: Estimated componentwise
         reciprocal condition number.  Compared with the threshold
         sqrt(n) * dlamch('Epsilon') to determine if the error
         estimate is 'guaranteed'. These reciprocal condition
         numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
         appropriately scaled matrix Z.
         Let Z = S*(A*diag(x)), where x is the solution for the
         current right-hand side and S scales each row of
         A*diag(x) by a power of the radix so all absolute row
         sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra cautions. 🧚

NPARAMS is INTEGER — specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. ✨
PARAMS is DOUBLE PRECISION array, dimension NPARAMS — specifies algorithm parameters. If an entry is < 0.0, then that entry gets filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.
- PARAMS(LA_LINRX_ITREF_I = 1): whether to perform iterative refinement or not.
  - Default: 1.0D+0
  - = 0.0: no refinement is performed, and no error bounds are computed.
  - = 1.0: use the extra-precise refinement algorithm.
  - (other values are reserved for future use)
- PARAMS(LA_LINRX_ITHRESH_I = 2): maximum number of residual computations allowed for refinement.
  - Default: 10
  - Aggressive: set to 100 to permit convergence using approximate factorizations or factorizations other than LU. If the factorization uses a technique other than Gaussian elimination, the guarantees in err_bnds_norm and err_bnds_comp may no longer be trustworthy.
- PARAMS(LA_LINRX_CWISE_I = 3): flag determining if the code will attempt to find a solution with small componentwise relative error in the double-precision algorithm. Positive is true, 0.0 is false.
  - Default: 1.0 (attempt componentwise convergence)
WORK is DOUBLE PRECISION array, dimension (4*N) — it's the workspace, bestie. 🛠️
IWORK is INTEGER array, dimension (N) — integer workspace, equally iconic.
INFO

INFO is INTEGER — here's the whole vibe check:

= 0: Successful exit. The solution to every right-hand side is guaranteed. 💯
< 0: If INFO = -i, the i-th argument had an illegal value. oop
> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND = 0 is returned. no cap, that's a problem
= N+J: The solution corresponding to the Jth right-hand side is not guaranteed. The solutions corresponding to other right-hand sides K with K > J may not be guaranteed as well, but only the first such right-hand side is reported. If a small componentwise error is not requested (PARAMS(3) = 0.0) then the Jth right-hand side is the first with a normwise error bound that is not guaranteed (the smallest J such that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth right-hand side is the first with either a normwise or componentwise error bound that is not guaranteed (the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0). See the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information about all of the right-hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.

👑 Author 👑

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

✨ subroutine `dsposv` ✨

   subroutine dsposv (character UPLO, integer N, integer NRHS, double
       precision, dimension( lda, * ) A, integer LDA, double precision,
       dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, *
       ) X, integer LDX, double precision, dimension( n, * ) WORK, real,
       dimension( * ) SWORK, integer ITER, integer INFO)

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

🎀 Purpose 🎀

Bestie, DSPOSV computes the solution to a real system of linear equations

   A * X = B,

where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices.

DSPOSV first attempts to factorize the matrix in SINGLE PRECISION and use this factorization within an iterative refinement procedure to produce a solution with DOUBLE PRECISION normwise backward error quality (see below). If the approach fails the method switches to a DOUBLE PRECISION factorization and solve.

The iterative refinement is not going to be a winning strategy if the ratio SINGLE PRECISION performance over DOUBLE PRECISION performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. This might be done with a call to ILAENV in the future. Up to now, we always try iterative refinement.

The iterative refinement process is stopped if

ITER > ITERMAX

or for all the RHS we have:

RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX

where

ITER is the number of the current iteration in the iterative refinement process
RNRM is the infinity-norm of the residual
XNRM is the infinity-norm of the solution
ANRM is the infinity-operator-norm of the matrix A
EPS is the machine epsilon returned by DLAMCH('Epsilon')

The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.

🌸 Parameters 🌸

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 matrix B. NRHS >= 0.
A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if iterative refinement has been successfully used (INFO = 0 and ITER >= 0, see description below), then A is unchanged, if double precision factorization has been used (INFO = 0 and ITER < 0, see description below), then the array A contains the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
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, the N-by-NRHS solution matrix X.
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
WORK — WORK is DOUBLE PRECISION array, dimension (N,NRHS). This array is used to hold the residual vectors.
SWORK — SWORK is REAL array, dimension (N*(N+NRHS)). This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.
ITER — ITER is INTEGER
- < 0: iterative refinement has failed, double precision factorization has been performed
  - -1: the routine fell back to full precision for implementation- or machine-specific reasons
  - -2: narrowing the precision induced an overflow, the routine fell back to full precision
  - -3: failure of SPOTRF
  - -31: stop the iterative refinement after the 30th iterations
- > 0: iterative refinement has been successfully used. Returns the number of iterations
INFO — INFO is INTEGER
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, the leading minor of order i of (DOUBLE PRECISION) A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

👑 Author 👑

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

Author Generated automatically by Doxygen for LAPACK from the source code.

That's the whole tea, bestie. This masterpiece is from LAPACK version 3.11.0, dated Sun Nov 27 2022, and it's literally the doublePOsolve(3) man page. Slay! 💅✨🦋

📦 where this page comes from

debian bookworm: liblapack-doc 3.11.0-2 · BSD-3-Clause
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

doublePOsolve(3)

💖 doublePOsolve(3) — LAPACK's PO solve squad 💖

🌸 NAME 🌸

✨ SYNOPSIS ✨

🎀 Functions 🎀

💫 Detailed Description 💫

👑 Function Documentation 👑

🧚 subroutine dposv 🧚

🎯 Purpose

🧾 Parameters

🫶 Author

🦋 subroutine dposvx 🦋

🎯 Purpose

📋 Description

🧾 Parameters

B

LDB

X

LDX

RCOND

FERR

BERR

WORK

IWORK

INFO

👑 Author

subroutine dposvxx

Purpose:

Description:

Parameters

FACT

UPLO

N

NRHS

A

LDA

AF

LDAF

EQUED

👑 Author 👑

✨ subroutine dsposv ✨

🎀 Purpose 🎀

🌸 Parameters 🌸

👑 Author 👑

🧚 subroutine `dposv` 🧚

🦋 subroutine `dposvx` 🦋

`B`

`LDB`

`X`

`LDX`

`RCOND`

`FERR`

`BERR`

`WORK`

`IWORK`

`INFO`

`subroutine dposvxx`

`FACT`

`UPLO`

`N`

`NRHS`

`A`

`LDA`

`AF`

`LDAF`

`EQUED`

✨ subroutine `dsposv` ✨