doublePOcomputational(3)

📦

version 3.11.0 · 2 versions

3.11.0 · debian bookworm (latest)
3.10.0 · ubuntu jammy

✨ NAME ✨

bestie, this is the LAPACK manual page for doublePOcomputational — and the official name line is literally doublePOcomputational - double. That's the whole name, no cap. It's the double precision computational squad for PO matrices. Periodt. 💅

✨ SYNOPSIS ✨

Functions

okay so here's the function lineup, and I am keeping every signature exactly as written because we respect the source code in this house. 🫶

dla_porcond — estimates the Skeel condition number for a symmetric positive-definite matrix.

       double precision function dla_porcond (UPLO, N, A, LDA, AF, LDAF, CMODE,
           C, INFO, WORK, IWORK)

dla_porfsx_extended — improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

       subroutine dla_porfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF,
           LDAF, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM,
           ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB,
           IGNORE_CWISE, INFO)

dla_porpvgrw — computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.

       double precision function dla_porpvgrw (UPLO, NCOLS, A, LDA, AF, LDAF,
           WORK)

dpocon — the synopsis is literally just DPOCON. That's the whole vibe. The details are in the function docs below.

       subroutine dpocon (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)

dpoequ — synopsis says DPOEQU. Short and sweet.

       subroutine dpoequ (N, A, LDA, S, SCOND, AMAX, INFO)

dpoequb — synopsis says DPOEQUB. Another iconic short one.

       subroutine dpoequb (N, A, LDA, S, SCOND, AMAX, INFO)

dporfs — synopsis says DPORFS.

       subroutine dporfs (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
           FERR, BERR, WORK, IWORK, INFO)

dporfsx — synopsis says DPORFSX.

       subroutine dporfsx (UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB,
           X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
           NPARAMS, PARAMS, WORK, IWORK, INFO)

dpotf2 — computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).

       subroutine dpotf2 (UPLO, N, A, LDA, INFO)

dpotrf — synopsis says DPOTRF.

       subroutine dpotrf (UPLO, N, A, LDA, INFO)

dpotrf2 — synopsis says DPOTRF2.

       recursive subroutine dpotrf2 (UPLO, N, A, LDA, INFO)

dpotri — synopsis says DPOTRI.

       subroutine dpotri (UPLO, N, A, LDA, INFO)

dpotrs — synopsis says DPOTRS.

       subroutine dpotrs (UPLO, N, NRHS, A, LDA, B, LDB, INFO)

✨ Detailed Description ✨

This is the group of double computational functions for PO matrices. That's the official tea. ☕️ So if you're doing double precision symmetric or Hermitian positive-definite stuff, you're in the right place. 💖

✨ Function Documentation ✨

`double precision function dla_porcond`

   double precision function dla_porcond (character UPLO, integer N, double
       precision, dimension( lda, * ) A, integer LDA, double precision,
       dimension( ldaf, * ) AF, integer LDAF, integer CMODE, double precision,
       dimension( * ) C, integer INFO, double precision, dimension( * ) WORK,
       integer, dimension( * ) IWORK)

DLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix. That's the one-liner. 🫶

Purpose

okay so the actual purpose, bestie:

               DLA_PORCOND Estimates the Skeel condition number of  op(A) * op2(C)
               where op2 is determined by CMODE as follows
               CMODE =  1    op2(C) = C
               CMODE =  0    op2(C) = I
               CMODE = -1    op2(C) = inv(C)
               The Skeel condition number  cond(A) = norminf( |inv(A)||A| )
               is computed by computing scaling factors R such that
               diag(R)*A*op2(C) is row equilibrated and computing the standard
               infinity-norm condition number.

So yeah, it's giving Skeel condition number, and the formula is literally in that block. Don't skip it. ✨

Parameters

UPLO — UPLO is CHARACTER*1. Tells you which triangle of A is stored:
- = '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.
A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is DOUBLE PRECISION array, dimension (LDAF,N). The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
CMODE — CMODE is INTEGER. Determines op2(C) in the formula op(A) * op2(C) as follows:
- CMODE = 1 op2(C) = C
- CMODE = 0 op2(C) = I
- CMODE = -1 op2(C) = inv(C)
C — C is DOUBLE PRECISION array, dimension (N). The vector C in the formula op(A) * op2(C).
INFO — INFO is INTEGER
- = 0: Successful exit.
- i > 0: The ith argument is invalid.
WORK — WORK is DOUBLE PRECISION array, dimension (3*N). Workspace.
IWORK — IWORK is INTEGER array, dimension (N). Workspace.

Author

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

`subroutine dla_porfsx_extended`

   subroutine dla_porfsx_extended (integer PREC_TYPE, character UPLO, integer
       N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA,
       double precision, dimension( ldaf, * ) AF, integer LDAF, logical COLEQU,
       double precision, dimension( * ) C, double precision, dimension( ldb, *
       ) B, integer LDB, double precision, dimension( ldy, * ) Y, integer LDY,
       double precision, dimension( * ) BERR_OUT, integer N_NORMS, double
       precision, dimension( nrhs, * ) ERR_BNDS_NORM, double precision,
       dimension( nrhs, * ) ERR_BNDS_COMP, double precision, dimension( * )
       RES, double precision, dimension(*) AYB, double precision, dimension( *
       ) DY, double precision, dimension( * ) Y_TAIL, double precision RCOND,
       integer ITHRESH, double precision RTHRESH, double precision DZ_UB,
       logical IGNORE_CWISE, integer INFO)

DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. Iconic. 💅

Purpose

            DLA_PORFSX_EXTENDED improves the computed solution to a system of
            linear equations by performing extra-precise iterative refinement
            and provides error bounds and backward error estimates for the solution.
            This subroutine is called by DPORFSX to perform iterative refinement.
            In addition to normwise error bound, the code provides maximum
            componentwise error bound if possible. See comments for ERR_BNDS_NORM
            and ERR_BNDS_COMP for details of the error bounds. Note that this
            subroutine is only responsible for setting the second fields of
            ERR_BNDS_NORM and ERR_BNDS_COMP.

Parameters

PREC_TYPE — PREC_TYPE is INTEGER. Specifies the intermediate precision to be used in refinement. The value is defined by ILAPREC(P) where P is a CHARACTER and P:
- = 'S': Single
- = 'D': Double
- = 'I': Indigenous
- = 'X' or 'E': Extra
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.
A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the N-by-N matrix A.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is DOUBLE PRECISION array, dimension (LDAF,N). The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF.
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
COLEQU — COLEQU is LOGICAL. If .TRUE. then column equilibration was done to A before calling this routine. This is needed to compute the solution and error bounds correctly.
C — C is DOUBLE PRECISION array, dimension (N). The column scale factors for A. If COLEQU = .FALSE., C is not accessed. If C is input, each element of C should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable.
B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). The right-hand-side matrix B.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
Y — Y is DOUBLE PRECISION array, dimension (LDY,NRHS). On entry, the solution matrix X, as computed by DPOTRS. On exit, the improved solution matrix Y.
LDY — LDY is INTEGER. The leading dimension of the array Y. LDY >= max(1,N).
BERR_OUT — BERR_OUT is DOUBLE PRECISION array, dimension (NRHS). On exit, BERR_OUT(j) contains the componentwise relative backward error for right-hand-side j from the formula:

max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) is the componentwise absolute value of the matrix or vector Z. This is computed by DLA_LIN_BERR.

N_NORMS — N_NORMS is INTEGER. Determines which error bounds to return (see ERR_BNDS_NORM and ERR_BNDS_COMP). If N_NORMS >= 1 return normwise error bounds. If N_NORMS >= 2 return componentwise error bounds.
ERR_BNDS_NORM — ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS). For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

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.

okay so back to ERR_BNDS_NORM — the second index in ERR_BNDS_NORM(:,err) holds these three fields, bestie:

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

This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions. 💅

`ERR_BNDS_COMP`

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

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 (on which the componentwise relative error depends), 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) * slamch('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) * slamch('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) * slamch('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.

This subroutine is only responsible for setting the second field above. See Lapack Working Note 165 for further details and extra cautions. 🫶

`RES`

RES is DOUBLE PRECISION array, dimension (N). Workspace to hold the intermediate residual.

`AYB`

AYB is DOUBLE PRECISION array, dimension (N). Workspace. This can be the same workspace passed for Y_TAIL.

`DY`

DY is DOUBLE PRECISION array, dimension (N). Workspace to hold the intermediate solution.

`Y_TAIL`

Y_TAIL is DOUBLE PRECISION array, dimension (N). Workspace to hold the trailing bits of the intermediate solution.

`RCOND`

RCOND is DOUBLE PRECISION. Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if 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.

`ITHRESH`

ITHRESH is INTEGER. The maximum number of residual computations allowed for refinement. The default is 10. For '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.

`RTHRESH`

RTHRESH is DOUBLE PRECISION. Determines when to stop refinement if the error estimate stops decreasing. Refinement will stop when the next solution no longer satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The default value is 0.5. For 'aggressive' set to 0.9 to permit convergence on extremely ill-conditioned matrices. See LAWN 165 for more details.

`DZ_UB`

DZ_UB is DOUBLE PRECISION. Determines when to start considering componentwise convergence. Componentwise convergence is only considered after each component of the solution Y is stable, which we define as the relative change in each component being less than DZ_UB. The default value is 0.25, requiring the first bit to be stable. See LAWN 165 for more details.

`IGNORE_CWISE`

IGNORE_CWISE is LOGICAL. If .TRUE. then ignore componentwise convergence. Default value is .FALSE..

`INFO`

INFO is INTEGER.

= 0: Successful exit.
< 0: if INFO = -i, the ith argument to DPOTRS had an illegal value.

✨ Author ✨

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

`dla_porpvgrw`

   double precision function dla_porpvgrw (character*1 UPLO, integer NCOLS,
       double precision, dimension( lda, * ) A, integer LDA, double precision,
       dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( * )
       WORK)

DLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.

Purpose:

DLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is used. If this is much less than 1, the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X, estimated condition numbers, and error bounds could be unreliable. 😭

Parameters

`UPLO`

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

`NCOLS`

NCOLS is INTEGER. The number of columns of the matrix A. NCOLS >= 0.

`A`

A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the N-by-N matrix A.

`LDA`

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

`AF`

AF is DOUBLE PRECISION array, dimension (LDAF,N). The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF.

`LDAF`

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

`WORK`

WORK is DOUBLE PRECISION array, dimension (2*N).

✨ Author ✨

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

`dpocon`

   subroutine dpocon (character UPLO, integer N, double precision, dimension(
       lda, * ) A, integer LDA, double precision ANORM, double precision RCOND,
       double precision, dimension( * ) WORK, integer, dimension( * ) IWORK,
       integer INFO)

DPOCON

Purpose:

DPOCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). 💅

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 order of the matrix A. N >= 0.

`A`

A is DOUBLE PRECISION array, dimension (LDA,N). The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF.

`LDA`

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

`ANORM`

ANORM is DOUBLE PRECISION. The 1-norm (or infinity-norm) of the symmetric matrix A.

`RCOND`

RCOND is DOUBLE PRECISION. The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine.

`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

✨ Author ✨

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

💖 DPOEQU 💖

subroutine dpoequ (integer N, double precision, dimension( lda, * ) A,
    integer LDA, double precision, dimension( * ) S, double precision SCOND,
    double precision AMAX, integer INFO)

✨ Purpose ✨

Okay so DPOEQU computes row and column scalings that are intended to equilibrate a symmetric positive definite matrix A, and it reduces the condition number (with respect to the two-norm, no cap). S holds the scale factors, and they're chosen like this:

S(i) = 1/sqrt(A(i,i))

so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. And the way this S is picked means the condition number of B stays within a factor N of the smallest possible condition number across all possible diagonal scalings. Obsessed with her honestly 💅

🌸 Parameters 🌸

N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is DOUBLE PRECISION array, dimension (LDA,N). The N-by-N symmetric positive definite matrix whose scaling factors are to be computed. Heads up bestie: only the diagonal elements of A are referenced.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
S — S is DOUBLE PRECISION array, dimension (N). If INFO = 0, S contains the scale factors for A.
SCOND — SCOND is DOUBLE PRECISION. If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). And lowkey important: if SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S.
AMAX — AMAX is DOUBLE PRECISION. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled. Like, actually should 👀
INFO — INFO is INTEGER.
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, the i-th diagonal element is nonpositive

🫶 Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 DPOEQUB 🦋

subroutine dpoequb (integer N, double precision, dimension( lda, * ) A,
    integer LDA, double precision, dimension( * ) S, double precision SCOND,
    double precision AMAX, integer INFO)

✨ Purpose ✨

So DPOEQUB also computes row and column scalings intended to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors,

S(i) = 1/sqrt(A(i,i))

chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. And this choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. Same energy as DPOEQU, but wait —

This routine differs from DPOEQU by restricting the scaling factors to a power of the radix. Barring over- and underflow, scaling by these factors introduces no additional rounding errors. However, the scaled diagonal entries are no longer approximately 1 but lie between sqrt(radix) and 1/sqrt(radix). Trade-offs, bestie, it's giving very balanced 💅

🌸 Parameters 🌸

N — N is INTEGER. The order of the matrix A. N >= 0.
A — A is DOUBLE PRECISION array, dimension (LDA,N). The N-by-N symmetric positive definite matrix whose scaling factors are to be computed. Only the diagonal elements of A are referenced.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
S — S is DOUBLE PRECISION array, dimension (N). If INFO = 0, S contains the scale factors for A.
SCOND — SCOND is DOUBLE PRECISION. If INFO = 0, S contains the ratio of the smallest S(i) to the largest S(i). If SCOND >= 0.1 and AMAX is neither too large nor too small, it is not worth scaling by S.
AMAX — AMAX is DOUBLE PRECISION. Absolute value of largest matrix element. If AMAX is very close to overflow or very close to underflow, the matrix should be scaled.
INFO — INFO is INTEGER.
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, the i-th diagonal element is nonpositive

🫶 Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🔥 DPORFS 🔥

subroutine dporfs (character UPLO, integer N, integer NRHS, double
    precision, dimension( lda, * ) A, integer LDA, double precision,
    dimension( ldaf, * ) AF, integer LDAF, double precision, dimension( ldb,
    * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer
    LDX, double precision, dimension( * ) FERR, double precision, dimension(
    * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * )
    IWORK, integer INFO)

✨ Purpose ✨

DPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and it hands you error bounds plus backward error estimates for the solution. Glow-up for your answer, fr fr 🌈

🌸 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 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). 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.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
AF — AF is DOUBLE PRECISION array, dimension (LDAF,N). The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. (Shoutout to DPOTRF, she did the work first 💅)
LDAF — LDAF is INTEGER. The leading dimension of the array AF. LDAF >= max(1,N).
B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). The 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). On entry, the solution matrix X, as computed by DPOTRS. On exit, the improved solution matrix X. The way she just gets better 😭
LDX — LDX is INTEGER. The leading dimension of the array X. LDX >= max(1,N).
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

🧚 Internal Parameters 🧚

ITMAX is the maximum number of steps of iterative refinement. (She's grinding, not rushing 💪)

🫶 Author 🫶

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

👑 DPORFSX 👑

subroutine dporfsx (character UPLO, character EQUED, integer N, integer
    NRHS, double precision, dimension( lda, * ) A, integer LDA, double
    precision, dimension( ldaf, * ) AF, integer LDAF, double precision,
    dimension( * ) S, double precision, dimension( ldb, * ) B, integer LDB,
    double precision, dimension( ldx, * ) X, integer LDX, double precision
    RCOND, 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)

✨ Purpose ✨

DPORFSX improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and provides error bounds and backward error estimates for the solution. But she's not like the other girls, because in addition to the normwise error bound, the code provides the maximum componentwise error bound if possible. See the comments for ERR_BNDS_NORM and ERR_BNDS_COMP for the deets on the error bounds 💖

The original system of linear equations may have been equilibrated before calling this routine, as described by arguments EQUED and S below. In that case, the solution and error bounds returned are for the original unequilibrated system. Not me getting the original vibe back 😌

Some optional parameters are bundled in the PARAMS array. These settings determine how refinement is performed, but often the defaults are acceptable. And if the defaults are acceptable, users can pass NPARAMS = 0, which prevents the source code from accessing the PARAMS argument. Iconic shortcut, lowkey genius ✨

🌸 Parameters 🌸

UPLO — UPLO is CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
EQUED — bestie hold that thought, this one continues in the next part 💅 (don't close the tab, we're literally mid-parameter)
EQUED — EQUED is CHARACTER*1 💅

Tells you what kind of equilibration got done to A before this routine got called. Bestie, this matters — we need it to compute the solution and the error bounds correctly. No guessing allowed.

= 'N': No equilibration at all, we're keeping it natural
= 'Y': Both row and column equilibration happened, aka A got replaced by diag(S) * A * diag(S). And the right hand side B has been changed accordingly. She's been through it. 😌

N — N is INTEGER

The order of the matrix A. N >= 0. Iconic, zero-indexed-adjacent queen.

NRHS — NRHS is INTEGER

The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. Yes bestie, you can solve for multiple besties at once.

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

The symmetric matrix A. Okay so here's the tea ☕ — 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 (we don't even look at her, she's not invited). 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. Pick a side and commit, fr fr.

LDA — LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N). Respect the boundary, she has limits.

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

The triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. It's giving factorization déjà vu, but that's the whole point. ✨

LDAF — LDAF is INTEGER

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

S — S is DOUBLE PRECISION array, dimension (N)

The scale factors for A. If EQUED = 'Y', A is multiplied on the left and right by diag(S).

Now listen carefully because this is important: 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. If S is output, each element of S is a power of the radix. If S is input, each element of S should be a power of the radix to ensure a reliable solution and error estimates. Scaling by powers of the radix does not cause rounding errors unless the result underflows or overflows. Rounding errors during scaling lead to refining with a matrix that is not equivalent to the input matrix, producing error estimates that may not be reliable. 😭

So like... powers of the radix or it's over for your error estimates. That's the vibe. Don't come for me, that's just what the math said.

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

The right hand side matrix B. The moment. The main character of the RHS world. 💖

LDB — LDB is INTEGER

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

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

On entry, the solution matrix X, as computed by DGETRS. On exit, the improved solution matrix X. Glow up era, she's better now. 🦋

LDX — LDX is INTEGER

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

RCOND — RCOND is DOUBLE PRECISION

Reciprocal scaled condition number. This is an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if 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 me warning you that tiny RCOND ≠ tiny error. Read the fine print, bestie, it could save you.

BERR — 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). Lowkey the most honest number in this whole page.

N_ERR_BNDS — 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 — ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the normwise relative error, which is defined as follows:

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').

Okay but literally the name of the field is "Trust/don't trust", this is the most dramatic documentation ever and I'm obsessed. 😤

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 — ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about various error bounds and condition numbers corresponding to the componentwise relative error, which is defined as follows:

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 (on which the componentwise relative error depends), 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. 💅

Wait, so what's the difference between the two? Normwise is like "how off is this whole vector as one vibe," componentwise is "how off is each individual element, judged separately." Both use the exact same three-field structure, both point you at dlamch('Epsilon') and Working Note 165. That's the pattern, bestie, once you see it you can't unsee it. 🧚

NPARAMS — NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used. She said "I don't want to hear it, just use the defaults." 🙈

PARAMS — PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)

Specifies algorithm parameters. If an entry is < 0.0, then that entry will be filled with 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 double-precision refinement algorithm, possibly with doubled-single computations if the compilation environment does not support DOUBLE PRECISION.
- (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.

🚨 Setting this to 100 is the "go big or go home" move. It lets you converge with factorizations other than LU, but if it's not Gaussian elimination, those guarantees in err_bnds_norm and err_bnds_comp might straight up stop being trustworthy. High risk, high reward, your call.

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 — WORK is DOUBLE PRECISION array, dimension (4*N)

She's the workspace. Give her her 4*N and let her cook. 🔥

IWORK
IWORK — INTEGER array, dimension (N). that's her, that's the whole vibe 💅
INFO — INTEGER, and bestie this one is carrying a lot of lore, so read it slow:
- = 0: successful exit ✨ and the solution to every right-hand side is guaranteed. we love a clean win.
- < 0: if INFO = -i, the i-th argument had an illegal value. somebody handed her nonsense.
- > 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. 😭
- = 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 dpotf2`

subroutine dpotf2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)

DPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm). no cap.

🌸 Purpose:

DPOTF2 computes the Cholesky factorization of a real symmetric positive definite matrix A.

the factorization has the form

   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 lower triangular.

this is the unblocked version of the algorithm, calling Level 2 BLAS. it's giving "no shortcuts, just vibes and loops". ✨

🎀 Parameters

UPLO — CHARACTER*1. specifies whether the upper or lower triangular part of the symmetric matrix A is stored.
- = 'U': upper triangular
- = 'L': lower triangular
N — INTEGER. the order of the matrix A. N >= 0.
A — 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 — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
INFO — INTEGER.
- = 0: successful exit 💖
- < 0: if INFO = -k, the k-th argument had an illegal value
- > 0: if INFO = k, the leading minor of order k is not positive definite, and the factorization could not be completed.

👑 Author

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

🌈 `subroutine dpotrf`

subroutine dpotrf (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)

DPOTRF DPOTRF VARIANT: top-looking block version of the algorithm, calling Level 3 BLAS. 🔥

🌸 Purpose:

DPOTRF computes the Cholesky factorization of a real symmetric positive definite matrix A.

the factorization has the form

   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 lower triangular.

this is the block version of the algorithm, calling Level 3 BLAS. bigger bites, faster tea. 💅

🎀 Parameters

UPLO — CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. the order of the matrix A. N >= 0.
A — 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 — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
INFO — 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 is not positive definite, and the factorization could not be completed.

👑 Author

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

okay and then she says it AGAIN, because the top-looking block version wanted a second runway walk. same Purpose, same Parameters, same Author, plus a Date. not me re-reading it twice — but you asked for everything, so here's everything. 🦋

🔁 Purpose:

DPOTRF computes the Cholesky factorization of a real symmetric positive definite matrix A.

the factorization has the form

   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 lower triangular.

this is the top-looking block version of the algorithm, calling Level 3 BLAS.

🎀 Parameters

UPLO — CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. the order of the matrix A. N >= 0.
A — 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 — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
INFO — 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 is not positive definite, and the factorization could not be completed.

👑 Author

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

🗓️ Date

December 2016

🧚 `recursive subroutine dpotrf2`

recursive subroutine dpotrf2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)

DPOTRF2 — yes she's recursive, yes she contains multitudes.

🌸 Purpose:

DPOTRF2 computes the Cholesky factorization of a real symmetric positive definite matrix A using the recursive algorithm.

the factorization has the form

   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 lower triangular.

this is the recursive version of the algorithm. it divides the matrix into four submatrices:

       [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
   A = [ -----|----- ]  with n1 = n/2
       [  A21 | A22  ]       n2 = n-n1

the subroutine calls itself to factor A11. update and scale A21 or A12, update A22 then calls itself to factor A22. she said "I'll do it myself" and meant it recursively. 😭💖

🎀 Parameters

UPLO — CHARACTER*1.
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER. the order of the matrix A. N >= 0.
A — 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 — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
INFO — 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 is not positive definite, and the factorization could not be completed.

👑 Author

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

🦋 `subroutine dpotri`

subroutine dpotri (character UPLO, integer N, double precision, dimension( lda, * ) A, integer LDA, integer INFO)

DPOTRI

🌸 Purpose:

DPOTRI computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF. she's not doing the work herself, she's building on what DPOTRF already slayed. 👑

🎀 Parameters

UPLO —
UPLO — it's a CHARACTER*1, bestie, and it's literally just telling you which triangle of A got stored:
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER, the order of the matrix A. The rule? N >= 0. No negative girlies allowed 💅
A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry it's holding the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, exactly as computed by DPOTRF. On exit? The upper or lower triangle of the (symmetric) inverse of A, overwriting the input factor U or L. She gets replaced, no cap 🦋
LDA — LDA is INTEGER, the leading dimension of the array A. Constraint: LDA >= max(1,N).
INFO — INFO is INTEGER, the vibe report:
- = 0: successful exit ✨
- < 0: if INFO = -i, the i-th argument had an illegal value 🚩
- > 0: if INFO = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed. It's giving "we tried, we cannot" 😭

🫶 Author 🫶

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

💜 subroutine `dpotrs` 💜

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

DPOTRS

🌸 Purpose 🌸

DPOTRS solves a system of linear equations A*X = B with a symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF. She takes the factorization and she solves, main character energy fr fr 👑

🎀 Parameters 🎀

UPLO — UPLO is CHARACTER*1, and it's the same triangle situation:
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER, the order of the matrix A. Again: N >= 0.
NRHS — NRHS is INTEGER, the number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. Like, how many problems are we solving at once? This many 💅
A — A is DOUBLE PRECISION array, dimension (LDA,N), holding the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by DPOTRF. No exit glow-up here — she stays as she is 🌷
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 right hand side matrix B. On exit, the solution matrix X. It's a glow-up arc, honestly 💖
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 🙈

🫶 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. Yeah bestie, the docs wrote themselves — iconic behaviour honestly 🤖💖

That's a wrap on this whole page! This is LAPACK Version 3.11.0, dated Sun Nov 27 2022, and that footer belongs to doublePOcomputational(3). Thanks for reading all six parts, you absolute legend 🎀💅✨

📦 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

doublePOcomputational(3)

✨ NAME ✨

✨ SYNOPSIS ✨

Functions

✨ Detailed Description ✨

✨ Function Documentation ✨

double precision function dla_porcond

Purpose

Parameters

Author

subroutine dla_porfsx_extended

Purpose

Parameters

ERR_BNDS_COMP

RES

AYB

DY

Y_TAIL

RCOND

ITHRESH

RTHRESH

DZ_UB

IGNORE_CWISE

INFO

✨ Author ✨

dla_porpvgrw

UPLO

NCOLS

A

LDA

AF

LDAF

WORK

✨ Author ✨

dpocon

UPLO

N

A

LDA

ANORM

RCOND

WORK

IWORK

INFO

✨ Author ✨

💖 DPOEQU 💖

✨ Purpose ✨

🌸 Parameters 🌸

🫶 Author 🫶

🦋 DPOEQUB 🦋

✨ Purpose ✨

🌸 Parameters 🌸

🫶 Author 🫶

🔥 DPORFS 🔥

✨ Purpose ✨

🌸 Parameters 🌸

🧚 Internal Parameters 🧚

🫶 Author 🫶

👑 DPORFSX 👑

✨ Purpose ✨

🌸 Parameters 🌸

👑 Author

🍓 subroutine dpotf2

🌸 Purpose:

🎀 Parameters

👑 Author

🌈 subroutine dpotrf

🌸 Purpose:

🎀 Parameters

👑 Author

🔁 Purpose:

🎀 Parameters

👑 Author

🗓️ Date

🧚 recursive subroutine dpotrf2

🌸 Purpose:

🎀 Parameters

👑 Author

🦋 subroutine dpotri

🌸 Purpose:

`double precision function dla_porcond`

`subroutine dla_porfsx_extended`

`ERR_BNDS_COMP`

`RES`

`AYB`

`DY`

`Y_TAIL`

`RCOND`

`ITHRESH`

`RTHRESH`

`DZ_UB`

`IGNORE_CWISE`

`INFO`

`dla_porpvgrw`

`UPLO`

`NCOLS`

`A`

`LDA`

`AF`

`LDAF`

`WORK`

`dpocon`

`UPLO`

`N`

`A`

`LDA`

`ANORM`

`RCOND`

`WORK`

`IWORK`

`INFO`

🍓 `subroutine dpotf2`

🌈 `subroutine dpotrf`

🧚 `recursive subroutine dpotrf2`

🦋 `subroutine dpotri`

💜 subroutine `dpotrs` 💜