complexPOsolve(3)

📦
version 3.11.0 · 2 versions
, in debian bookworm

complexPOsolve(3) — LAPACK's complex PO solve besties 💖✨

Okay bestie, welcome to the complex PO solve corner of LAPACK. The manual name is complexPOsolve(3), and it's literally the group of complex solve driver functions for PO matrices. No cap. 🫶

NAME 🎀

complexPOsolve - complex

It's giving "complex" and that's the whole NAME, periodt. 💅

SYNOPSIS ✨

Functions 💖

  • cposv — computes the solution to system of linear equations A * X = B for PO matrices
subroutine cposv (UPLO, N, NRHS, A, LDA, B, LDB, INFO)
  • cposvx — computes the solution to system of linear equations A * X = B for PO matrices
subroutine cposvx (FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B,
    LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
  • cposvxx — computes the solution to system of linear equations A * X = B for PO matrices
subroutine cposvxx (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, RWORK, INFO)

Detailed Description 💖

This is the group of complex solve driver functions for PO matrices. That's the whole vibe. 🌸

Function Documentation 📚

subroutine cposv — the main character 💅

subroutine cposv (character UPLO, integer N, integer NRHS, complex,
    dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B,
    integer LDB, integer INFO)

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

Purpose 🎯

CPOSV computes the solution to a complex system of linear equations

A * X = B,

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

The Cholesky decomposition is used to factor A as

A = U**H* U,  if UPLO = 'U', or
A = L * L**H,  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 COMPLEX array, dimension (LDA,N). On entry, the Hermitian 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**H*U or A = L*L**H.
  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • B — B is COMPLEX 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 cposvx — the one with extra ✨

subroutine cposvx (character FACT, character UPLO, integer N, integer NRHS,
    complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldaf, *
    ) AF, integer LDAF, character EQUED, real, dimension( * ) S, complex,
    dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X,
    integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * )
    BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer
    INFO)

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

Purpose 🎯

CPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N Hermitian positive definite matrix 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:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B

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

  1. If FACT = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = U**H* U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',

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

  1. 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.
  2. The system of equations is solved for X using the factored form of A.
  3. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
  4. 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 COMPLEX array, dimension (LDA,N). On entry, the Hermitian 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 COMPLEX 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**H*U or A = L*L**H, 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**H*U or A = L*L**H 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**H*U or A = L*L**H 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 REAL array, dimension (N). The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive.
  • B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the N-by-NRHS righthand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B.
  • LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • X
  • X — COMPLEX array, dimension (LDX,NRHS). If INFO = 0 or INFO = N+1, this is literally the N-by-NRHS solution matrix X for the original system of equations. Heads up bestie: if EQUED = 'Y', then A and B get modified on exit, and the solution to the equilibrated system is inv(diag(S))*X. so the X you're holding is not always the X you think you're holding 💅
  • LDX — INTEGER. The leading dimension of the array X. Constraint: LDX >= max(1,N). respect the leading dimension or perish 🙈
  • RCOND — REAL. The estimate of the reciprocal condition number of the matrix A after equilibration (if that even happened). If RCOND is less than the machine precision — and especially if RCOND = 0 — then the matrix is singular to working precision. That whole situation gets flagged by a return code of INFO > 0. she's giving "not invertible, sorry babes" 💀
  • FERR — 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), 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). This estimate is exactly as reliable as the estimate for RCOND, and it's almost always a slight overestimate of the true error. honest queen, she overshoots a lil but she's not lying to you 🫶
  • BERR — 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. iconic little "how close were we really" metric ✨
  • WORK — COMPLEX array, dimension (2*N)
  • RWORK — REAL array, dimension (N)
  • INFO — INTEGER. The receipts, in order:
    • = 0: successful exit. ate, no notes 👑
    • < 0: if INFO = -i, the i-th argument had an illegal value. you handed her something cursed 🚩
    • > 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. no solution for you, the matrix said no 😭
      • = 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. she's technically singular but she's still showing up and doing the work?? main character energy fr fr 🌸

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

✨ subroutine cposvxx ✨

okay new function new me, this one is the ✨fancy✨ version 💖

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

CPOSVXX computes the solution to system of linear equations A * X = B for PO matrices slay 💅✨

🌈 Purpose:

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

A * X = B

where A is an N-by-N Hermitian positive definite matrix and X and B are N-by-NRHS matrices. Hermitian positive definite besties stay winning 💜

If requested, both normwise and maximum componentwise error bounds are returned. CPOSVXX 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. she gives you receipts AND a warning label, that's called being thorough 🎀

CPOSVXX 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 CPOSVXX 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 CPOSVXX would itself produce. translation: if you hand her your own weird factorization and it's not what she'd have made herself, all bets are off, no cap 🙈

🦋 Description:

The following steps are performed:

  1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B

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

  1. 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.

  1. 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. she does NOT give up, she just tells you it's bad and keeps going 😭👑
  2. The system of equations is solved for X using the factored form of A.
  3. 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. double precision residuals?? she's doing the most and we love her for it ✨
  4. 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. defaults are usually fine bestie, just pass 0 and vibe 🌸

💖 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. she's just reading, not touching 💅
    • = '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 COMPLEX array, dimension (LDA,N). On entry, the Hermitian 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). yes she really does overwrite it, say goodbye to your original A 😭

  • LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • AF — AF is COMPLEX 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). she's a switch, sometimes input sometimes output, we respect it 🫶
  • 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': Both row and column equilibration, 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. context-dependent queen 💜

  • S —

S

<strong><code>S</code></strong> is a REAL array, dimension (N) — and it's giving row scale factors for A 💅

If EQUED = 'Y', A gets multiplied on the left and on the right by diag(S). Whether S is an input or an output is entirely a FACT situation: S is an input argument if FACT = 'F', otherwise S is an output argument. And heads up — if FACT = 'F' and EQUED = 'Y', then every single element of S must be positive, no exceptions 🙅‍♀️

If S is an output, each element of S is a power of the radix. If S is an input, each element of S should be a power of the radix so your solution and your error estimates stay reliable. The way scaling by powers of the radix works is that it does not cause rounding errors — unless the result underflows or overflows, and then 😭. And rounding errors during scaling mean you're refining with a matrix that is not equivalent to the input matrix, which gives you error estimates that may not be reliable. So like. Be normal about it. Use powers of the radix.

B

<strong><code>B</code></strong> is a COMPLEX array, dimension (LDB,NRHS) 🦋

On entry, it's the N-by-NRHS right hand side matrix B. On exit, it depends on the vibe:

  • if EQUED = 'N', B is not modified — she's untouched, she's safe ✨
  • if EQUED = 'Y', B is overwritten by diag(S)*B — she's been through something 💀

LDB

<strong><code>LDB</code></strong> is INTEGER — the leading dimension of the array B.

And it has one job: LDB >= max(1,N). That's it. That's the whole job. Periodt.

X

<strong><code>X</code></strong> is a COMPLEX array, dimension (LDX,NRHS) 👑

If INFO = 0, this is the N-by-NRHS solution matrix X to the original system of equations. Main character energy, honestly.

But bestie, real talk: 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. So X is the answer to the original system, but if equilibration happened you need that extra multiplication to get the answer to the equilibrated one. Two different questions, two different answers, iykyk 🎀

LDX

<strong><code>LDX</code></strong> is INTEGER — the leading dimension of the array X.

And, same as her sister: LDX >= max(1,N). Simple. Iconic.

RCOND

<strong><code>RCOND</code></strong> is REAL — the reciprocal scaled condition number 💫

It's an estimate of the reciprocal Skeel condition number of the matrix A after equilibration (if that happened). If this number is less than the machine precision — and especially if it's zero — then the matrix is singular to working precision. Yes, that's as bad as it sounds.

But here's the plot twist 🌈: the error may still be small even if this number is very very small and the matrix appears ill-conditioned. Appearance is not identity, bestie.

RPVGRW

<strong><code>RPVGRW</code></strong> is REAL — reciprocal pivot growth 🍓

On exit, this contains the reciprocal pivot growth factor norm(A)/norm(U). The 'max absolute element' norm is the one being used here.

If this is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. And when that happens? The solution X, the estimated condition numbers, and the error bounds could all be unreliable. Everything. All of it. 💀

Special case: if factorization fails with 0<INFO<=N, then RPVGRW contains the reciprocal pivot growth factor for the leading INFO columns of A.

BERR

<strong><code>BERR</code></strong> is a REAL array, dimension (NRHS) — the componentwise relative backward error 🌸

This is the componentwise relative backward error of each solution vector X(j). In other words: the smallest relative change in any element of A or B that would make X(j) an exact solution. Tiny number means she's basically already right.

N_ERR_BNDS

<strong><code>N_ERR_BNDS</code></strong> is INTEGER — the number of error bounds to return for each right hand side and each type (normwise or componentwise).

Peep ERR_BNDS_NORM and ERR_BNDS_COMP below for the actual goods 👇

ERR_BNDS_NORM

<strong><code>ERR_BNDS_NORM</code></strong> is a REAL 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:

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) * 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.

See Lapack Working Note 165 for further details and extra cautions 📖

ERR_BNDS_COMP

<strong><code>ERR_BNDS_COMP</code></strong> is a REAL 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, which is defined like this:

        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.

Important conditions, bestie 🫶:

  • 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.

See Lapack Working Note 165 for further details and extra cautions 📖

NPARAMS

<strong><code>NPARAMS</code></strong> is INTEGER — it specifies the number of parameters set in PARAMS.

If NPARAMS <= 0, the PARAMS array is never referenced and default values are used. She simply does not want to hear from you 💅

PARAMS

<strong><code>PARAMS</code></strong> is a REAL array, dimension NPARAMS — and this is where you specify algorithm parameters 🎀

If an entry is < 0.0, then that entry will be filled with the default value used for that parameter. Only positions up to NPARAMS are accessed; defaults are used for higher-numbered parameters.

The lineup:

  • PARAMS(LA_LINRX_ITREF_I = 1) — Whether to perform iterative refinement or not.
    • Default: 1.0
    • = 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. ⚠️
  • 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

<strong><code>WORK</code></strong> is a COMPLEX array, dimension (2*N) ✨

RWORK

<strong><code>RWORK</code></strong> is a REAL array, dimension (2*N) ✨

INFO

<strong><code>INFO</code></strong>

  • INFO is INTEGER — bestie, this is the receipts section, read it slow:
    • = 0 — slay! Successful exit. The solution to every right-hand side is guaranteed. 💖
    • < 0 — oopsie. If INFO = -i, the i-th argument had an illegal value. 🙈
    • > 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. 💀 Not the vibe.
    • = N+J — the solution corresponding to the Jth right-hand side is not guaranteed. The solutions for the other right-hand sides K with K > J may not be guaranteed either, but only the first such right-hand side gets reported. One at a time, we're not doing a full trauma dump here.

So like, if a small componentwise error is not requested (PARAMS(3) = 0.0), then the Jth right-hand side is the first one with a normwise error bound that isn't 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 one with either a normwise or a componentwise error bound that isn't guaranteed — the smallest J such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1) = 0.0.

Go read the definition of ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1) for the full lore. And if you want the tea on 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.

Author (the auto-generated one) ✨

Generated automatically by Doxygen for LAPACK from the source code. (yes bestie, the robots wrote this part, we just gave it main character energy 💅)

And that's the whole page, no crumbs left. 🌸 This one's from LAPACK Version 3.11.0, dated Sun Nov 27 2022, and you found it filed under complexPOsolve(3). Go forth and solve those Hermitian positive definite systems, you absolute icon. 🎀🦋 periodt.

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