Estimates the reciprocal of the condition number of a packed symmetric matrix.
FORTRAN 77:
call sspcon( uplo, n, ap, ipiv, anorm, rcond, work, iwork, info )
call dspcon( uplo, n, ap, ipiv, anorm, rcond, work, iwork, info )
call cspcon( uplo, n, ap, ipiv, anorm, rcond, work, info )
call zspcon( uplo, n, ap, ipiv, anorm, rcond, work, info )
FORTRAN 95:
call spcon( ap, ipiv, anorm, rcond [,uplo] [,info] )
C:
lapack_int LAPACKE_sspcon( int matrix_order, char uplo, lapack_int n, const float* ap, const lapack_int* ipiv, float anorm, float* rcond );
lapack_int LAPACKE_dspcon( int matrix_order, char uplo, lapack_int n, const double* ap, const lapack_int* ipiv, double anorm, double* rcond );
lapack_int LAPACKE_cspcon( int matrix_order, char uplo, lapack_int n, const lapack_complex_float* ap, const lapack_int* ipiv, float anorm, float* rcond );
lapack_int LAPACKE_zspcon( int matrix_order, char uplo, lapack_int n, const lapack_complex_double* ap, const lapack_int* ipiv, double anorm, double* rcond );
The routine estimates the reciprocal of the condition number of a packed symmetric matrix A:
κ1(A) = ||A||1 ||A-1||1 (since A is symmetric, κ∞(A) = κ1(A)).
Before calling this routine:
compute anorm (either ||A||1 = maxj Σi |aij| or ||A||∞ = maxi Σj |aij|)
call ?sptrf to compute the factorization of A.
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.
uplo |
CHARACTER*1. Must be 'U' or 'L'. Indicates how the input matrix A has been factored: If uplo = 'U', the array ap stores the packed upper triangular factor U of the factorization A = P*U*D*UT*PT. If uplo = 'L', the array ap stores the packed lower triangular factor L of the factorization A = P*L*D*LT*PT. |
n |
INTEGER. The order of matrix A; n ≥ 0. |
ap, work |
REAL for sspcon DOUBLE PRECISION for dspcon COMPLEX for cspcon DOUBLE COMPLEX for zspcon. Arrays: ap(*), work(*). The array ap contains the packed factored matrix A, as returned by ?sptrf. The dimension of ap must be at least max(1,n(n+1)/2). The array work is a workspace for the routine. The dimension of work must be at least max(1, 2*n). |
ipiv |
INTEGER. Array, DIMENSION at least max(1, n). The array ipiv, as returned by ?sptrf. |
anorm |
REAL for single precision flavors. DOUBLE PRECISION for double precision flavors. The norm of the original matrix A (see Description). |
iwork |
INTEGER. Workspace array, DIMENSION at least max(1, n). |
rcond |
REAL for single precision flavors. DOUBLE PRECISION for double precision flavors. An estimate of the reciprocal of the condition number. The routine sets rcond = 0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime rcond is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular. |
info |
INTEGER. If info = 0, the execution is successful. If info = -i, the i-th parameter had an illegal value. |
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 spcon interface are as follows:
ap |
Holds the array A of size (n*(n+1)/2). |
ipiv |
Holds the vector of length n. |
uplo |
Must be 'U' or 'L'. The default value is 'U'. |
The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10r. A call to this routine involves solving a number of systems of linear equations A*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2 floating-point operations for real flavors and 8n2 for complex flavors.