SRC/csysvxx.f(3)

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version 3.12.1 · 7 versions
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💅 SRC/csysvxx.f 💅

Library Functions Manual — bestie, this is the Unix manual page for SRC/csysvxx.f, and it's giving major linear-algebra main character energy. No cap. 💖

🎀 NAME 🎀

SRC/csysvxx.f — literally the page name, periodt. 💅

🌸 SYNOPSIS 🌸

🦋 Functions/Subroutines 🦋

       subroutine csysvxx (fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, equed,
           s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm,
           err_bnds_comp, nparams, params, work, rwork, info)

CSYSVXX computes the solution to system of linear equations A * X = B for SY matrices

✨ Function/Subroutine Documentation ✨

   subroutine csysvxx (character fact, character uplo, integer n, integer nrhs,
       complex, dimension( lda, * ) a, integer lda, complex, dimension( ldaf, *
       ) af, integer ldaf, integer, dimension( * ) ipiv, 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)

CSYSVXX computes the solution to system of linear equations A * X = B for SY matrices

💜 Purpose 💜

Okay so CSYSVXX uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. 💅

If requested, both normwise and maximum componentwise error bounds are returned. CSYSVXX 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. ✨

CSYSVXX accepts user-provided factorizations and equilibration factors; peep the definitions of the FACT and EQUED options. Solving with refinement and using a factorization from a previous CSYSVXX 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 CSYSVXX would itself produce. 🫶

💖 Description 💖

The following steps are performed, bestie:

  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 LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = U * D * U**T,  if UPLO = 'U', or
A = L * D * L**T,  if UPLO = 'L',

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

  1. If some D(i,i)=0, so that D is exactly singular, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A (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.
  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.
  4. If equilibration was used, the matrix X is premultiplied by diag(R) 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. 🎀

🌈 Parameters 🌈

FACT

Type check: FACT is CHARACTER*1. It specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. The options are:

  • 'F' — On entry, AF and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by S. A, AF, and IPIV 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

Type check: 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). 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 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 block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSYTRF.

If FACT = 'N', then AF is an output argument and on exit returns the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T.

LDAF

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

IPIV

IPIV is INTEGER array, dimension (N).

  • If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by CSYTRF. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by CSYTRF.

EQUED

Type check: EQUED is CHARACTER*1. It 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.

S

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

B

B is COMPLEX array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B.

On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S)*B.

LDB

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

X

X is COMPLEX array, dimension (LDX,NRHS). If INFO = 0, 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

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

RCOND

  • RCOND — REAL. It's the reciprocal scaled condition number, aka a vibe check on how close A is to being a whole mess 💀. It estimates the reciprocal Skeel condition number of A after equilibration (if equilibration happened). If it's less than machine precision — and especially if it's zero — the matrix is singular to working precision. But babes, listen: the error can still be small even when this number is tiny and the matrix looks ill-conditioned. Don't panic-read it.
  • RPVGRW — REAL. Reciprocal pivot growth, i.e. the growth factor but flipped: norm(A)/norm(U). The norm is used. If this is way less than 1, then the LU factorization of the (equilibrated) matrix A could be straight-up unstable 😭. Which also means your solution X, your estimated condition numbers and your error bounds could all be unreliable — heartbreaking, I know. If factorization fails with 0<INFO<=N, then this holds the reciprocal pivot growth factor for the leading INFO columns of A.
  • BERR — REAL array, dimension (NRHS). Componentwise relative backward error. For each solution vector X(j), this is the smallest relative change in any element of A or B that would make X(j) an exact solution. So it's like, how much do we gotta fudge things for this to be perfect ✨.
  • N_ERR_BNDS — INTEGER. How many error bounds to return for each right hand side and each type (normwise or componentwise). Peep ERR_BNDS_NORM and ERR_BNDS_COMP below 👇.
  • ERR_BNDS_NORM — REAL array, dimension (NRHS, N_ERR_BNDS). For every right-hand side, this array is absolutely packed with info about various error bounds and condition numbers for the normwise relative error, which goes 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 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — 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. Bestie, that's a whole chain of trust, respect it.
  • err = 3 — Reciprocal condition number: the estimated normwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is . 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 — REAL array, dimension (NRHS, N_ERR_BNDS). Same situation but for componentwise relative error, king 💅. For each right-hand side, this array holds info about various error bounds and condition numbers corresponding to the componentwise relative error, defined like:
     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 — she's not even invited to the function. 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 — boolean. Trust the answer if the reciprocal condition number is less than the threshold sqrt(n) * slamch('Epsilon').
  • err = 2 — 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. Same deal as above, no cap.
  • err = 3 — Reciprocal condition number: the estimated componentwise reciprocal condition number. Compared with the threshold sqrt(n) * slamch('Epsilon') to determine if the error estimate is . 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 — INTEGER. Tells you how many parameters are actually set in PARAMS. If it's <= 0, the PARAMS array is never referenced and default values are used. She said don't look at me, I'm not here 🙈.
  • PARAMS — REAL array, dimension NPARAMS. The algorithm tuning knobs, the main character settings 👑. 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 do 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. Warning though: 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 💀. That's a real one, don't play with it.
    • 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 — COMPLEX array, dimension (2*N). Scratch space, she works hard so you don't have to ✨.
  • RWORK — REAL array, dimension (2*N). The real-valued scratch space, same energy 🎀.
  • INFO — INTEGER. The whole story of how it went:
    • = 0: Successful exit. The solution to every right-hand side is guaranteed. We love a clean exit 💖.
    • < 0: 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. Tragic, but at least she told you 😭.
    • = 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. She's only reporting the first problem, you gotta go digging for the rest 🔍.

👑 Author

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

Definition at line 505 of file csysvxx.f.

🌈 Author

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