Uses the diagonal pivoting factorization to compute the solution to the system of linear equations with a real or complex symmetric matrix A, and provides error bounds on the solution.
FORTRAN 77:
call ssysvx( fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, iwork, info )
call dsysvx( fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, iwork, info )
call csysvx( fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info )
call zsysvx( fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info )
FORTRAN 95:
call sysvx( a, b, x [,uplo] [,af] [,ipiv] [,fact] [,ferr] [,berr] [,rcond] [,info] )
C:
lapack_int LAPACKE_ssysvx( int matrix_order, char fact, char uplo, lapack_int n, lapack_int nrhs, const float* a, lapack_int lda, float* af, lapack_int ldaf, lapack_int* ipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );
lapack_int LAPACKE_dsysvx( int matrix_order, char fact, char uplo, lapack_int n, lapack_int nrhs, const double* a, lapack_int lda, double* af, lapack_int ldaf, lapack_int* ipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );
lapack_int LAPACKE_csysvx( int matrix_order, char fact, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_float* a, lapack_int lda, lapack_complex_float* af, lapack_int ldaf, lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );
lapack_int LAPACKE_zsysvx( int matrix_order, char fact, char uplo, lapack_int n, lapack_int nrhs, const lapack_complex_double* a, lapack_int lda, lapack_complex_double* af, lapack_int ldaf, lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );
The routine uses the diagonal pivoting factorization to compute the solution to a real or complex system of linear equations A*X = B, where A is a n-by-n symmetric matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.
Error bounds on the solution and a condition estimate are also provided.
The routine ?sysvx performs the following steps:
If fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the factorization is A = U*D*UT or A = L*D*LT, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
If some di,i= 0, so that D is exactly singular, then the routine returns with info = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, info = n+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.
fact |
CHARACTER*1. Must be 'F' or 'N'. Specifies whether or not the factored form of the matrix A has been supplied on entry. If fact = 'F': on entry, af and ipiv contain the factored form of A. Arrays a, af, and ipiv will not be modified. If fact = 'N', the matrix A will be copied to af and factored. |
uplo |
CHARACTER*1. Must be 'U' or 'L'. Indicates whether the upper or lower triangular part of A is stored: If uplo = 'U', the upper triangle of A is stored. If uplo = 'L', the lower triangle of A is stored. |
n |
INTEGER. The order of matrix A; n ≥ 0. |
nrhs |
INTEGER. The number of right-hand sides, the number of columns in B; nrhs ≥ 0. |
a, af, b, work |
REAL for ssysvx DOUBLE PRECISION for dsysvx COMPLEX for csysvx DOUBLE COMPLEX for zsysvx. Arrays: a(lda,*), af(ldaf,*), b(ldb,*), work(*). The array a contains the upper or the lower triangular part of the symmetric matrix A (see uplo). The second dimension of a must be at least max(1,n). The array af is an input argument if fact = 'F'. It contains he block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T orA = L*D*L**T as computed by ?sytrf. The second dimension of af must be at least max(1,n). The array b contains the matrix B whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least max(1, nrhs). work(*) is a workspace array, dimension at least max(1,lwork). |
lda |
INTEGER. The leading dimension of a; lda ≥ max(1, n). |
ldaf |
INTEGER. The leading dimension of af; ldaf ≥ max(1, n). |
ldb |
INTEGER. The leading dimension of b; ldb ≥ max(1, n). |
ipiv |
INTEGER. Array, DIMENSION at least max(1, n). The array ipiv is an input argument if fact = 'F'. It contains details of the interchanges and the block structure of D, as determined by ?sytrf. If ipiv(i) = k > 0, then dii is a 1-by-1 diagonal block, and the i-th row and column of A was interchanged with the k-th row and column. If uplo = 'U' and ipiv(i) = ipiv(i-1) = -m < 0, then D has a 2-by-2 block in rows/columns i and i-1, and (i-1)-th row and column of A was interchanged with the m-th row and column. If uplo = 'L' and ipiv(i) = ipiv(i+1) = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and (i+1)-th row and column of A was interchanged with the m-th row and column. |
ldx |
INTEGER. The leading dimension of the output array x; ldx ≥ max(1, n). |
lwork |
INTEGER. The size of the work array. If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla. See Application Notes below for details and for the suggested value of lwork. |
iwork |
INTEGER. Workspace array, DIMENSION at least max(1, n); used in real flavors only. |
rwork |
REAL for csysvx; DOUBLE PRECISION for zsysvx. Workspace array, DIMENSION at least max(1, n); used in complex flavors only. |
x |
REAL for ssysvx DOUBLE PRECISION for dsysvx COMPLEX for csysvx DOUBLE COMPLEX for zsysvx. Array, DIMENSION (ldx,*). If info = 0 or info = n+1, the array x contains the solution matrix X to the system of equations. The second dimension of x must be at least max(1,nrhs). |
af, ipiv |
These arrays are output arguments if fact = 'N'. See the description of af, ipiv in Input Arguments section. |
rcond |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. An estimate of the reciprocal condition number of the matrix A. If rcond is less than the machine precision (in particular, if rcond = 0), the matrix is singular to working precision. This condition is indicated by a return code of info > 0. |
ferr |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, DIMENSION at least max(1, nrhs). Contains 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 |
REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, DIMENSION at least max(1, nrhs). Contains the component-wise relative backward error for each solution vector x(j), that is, the smallest relative change in any element of A or B that makes x(j) an exact solution. |
work(1) |
If info=0, on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs. |
info |
INTEGER. If info = 0, the execution is successful. If info = -i, the i-th parameter had an illegal value. If info = i, and i ≤ n, then dii is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned. If info = i, and i = n + 1, then D is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest. |
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.
Specific details for the routine sysvx interface are as follows:
a |
Holds the matrix A of size (n,n). |
b |
Holds the matrix B of size (n,nrhs). |
x |
Holds the matrix X of size (n,nrhs). |
af |
Holds the matrix AF of size (n,n). |
ipiv |
Holds the vector of length n. |
ferr |
Holds the vector of length (nrhs). |
berr |
Holds the vector of length (nrhs). |
uplo |
Must be 'U' or 'L'. The default value is 'U'. |
fact |
Must be 'N' or 'F'. The default value is 'N'. If fact = 'F', then both arguments af and ipiv must be present; otherwise, an error is returned. |
The value of lwork must be at least max(1,m*n), where for real flavors m = 3 and for complex flavors m = 2. For better performance, try using lwork = max(1, m*n, n*blocksize), where blocksize is the optimal block size for ?sytrf.
If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.
If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.
If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.
Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.