Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?spcon
Estimates the reciprocal of the condition number of a packed symmetric matrix.
Syntax
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 )
call spcon ( ap , ipiv , anorm , rcond [ , uplo ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
sspcon dspcon cspcon zspcon spcon
The routine estimates the reciprocal of the condition number of a packed symmetric matrix A :
κ_{1}(A) = ||A||_{1} ||A^{-1}||_{1} (since A is symmetric, \(\kappa_{\infty}(A) = \kappa_{1}(A)\) ).
An estimate is obtained for ||A^{-1}|| , and the reciprocal of the condition number is computed as rcond = 1 / (||A|| ||A^{-1}||) .
Before calling this routine:
compute anorm (either || A || 1 = max j Σ i | aij | or \(||A||_{\infty} = max_{i}\sum_{j} |a_{ij}|\) )
call ?sptrf (Computes the Bunch-Kaufman factorization of a symmetric matrix using packed storage.) to compute the factorization of A .
Input Parameters
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 = U*D*U^{T} .
If uplo = 'L' , the array ap stores the packed lower triangular factor L of the factorization A = L*D*L^{T} .
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 (Computes the Bunch-Kaufman factorization of a symmetric matrix using packed storage.) . 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, size at least max(1, n) .
The array ipiv , as returned by ?sptrf (Computes the Bunch-Kaufman factorization of a symmetric matrix using packed storage.) .
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, size at least max(1, n) .
Output Parameters
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.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
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 LAPACK 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’ .
Application Notes
The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10 r . 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 \(2n^{2}\) floating-point operations for real flavors and \(8n^{2}\) for complex flavors.