Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?sptrs
Solves a system of linear equations with a UDU- or LDL-factored symmetric coefficient matrix using packed storage.
Syntax
call ssptrs ( uplo , n , nrhs , ap , ipiv , b , ldb , info )
call dsptrs ( uplo , n , nrhs , ap , ipiv , b , ldb , info )
call csptrs ( uplo , n , nrhs , ap , ipiv , b , ldb , info )
call zsptrs ( uplo , n , nrhs , ap , ipiv , b , ldb , info )
call sptrs ( ap , b , ipiv [ , uplo ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
ssptrs dsptrs csptrs zsptrs sptrs
The routine solves for X the system of linear equations A*X = B with a symmetric matrix A , given the Bunch-Kaufman factorization of A :
if uplo='U' ,
A = U*D*U^{T}
if uplo='L' ,
A = L*D*L^{T} ,
where U and L are upper and lower packed triangular matrices with unit diagonal and D is a symmetric block-diagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the matrix B . You must supply the factor U (or L ) and the array ipiv returned by the factorization routine ?sptrf (Computes the Bunch-Kaufman factorization of a symmetric matrix using packed storage.) .
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 factor U of the factorization A = U*D*U^{T} . If uplo = 'L' , the array ap stores the packed factor L of the factorization A = L*D*L^{T} .
n
INTEGER . The order of matrix A ; n ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
ipiv
INTEGER .
Array, size at least max(1, n) . The ipiv array, as returned by ?sptrf (Computes the Bunch-Kaufman factorization of a symmetric matrix using packed storage.) .
ap
REAL for ssptrs
DOUBLE PRECISION for dsptrs
COMPLEX for csptrs
DOUBLE COMPLEX for zsptrs .
The dimension of array ap must be at least max(1, n(n+1)/2). The array ap contains the factor U or L , as specified by uplo , in packed storage (see Matrix Storage Schemes ).
b
REAL for ssptrs
DOUBLE PRECISION for dsptrs
COMPLEX for csptrs
DOUBLE COMPLEX for zsptrs .
The array b(ldb,*) contains the matrix B whose columns are the right-hand sides for the system of equations. The second dimension of b must be at least max(1, nrhs) .
ldb
INTEGER . The leading dimension of b ; ldb≥ max(1, n) .
Output Parameters
b
Overwritten by the solution matrix X .
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 sptrs interface are as follows:
ap
Holds the array A of size (n*(n+1)/2) .
b
Holds the matrix B of size ( n , nrhs ).
ipiv
Holds the vector of length n .
uplo
Must be ‘U’ or ‘L’ . The default value is ‘U’ .
Application Notes
For each right-hand side \(b\) , the computed solution is the exact solution of a perturbed system of equations \((A + E)x = b\) , where
\[|E| \leq c(n) \varepsilon P|U||D||U^{H}|P^{T} or |E| \le c(n) \varepsilon P|L||D||L^{H}|P^{T}\]
\(c(n)\) is a modest linear function of \(k\) , and \(\varepsilon\) is the machine precision.
If \(x_{0}\) is the true solution, the computed solution \(x\) satisfies this error bound:
\[\frac{\|x - x_0\|_\infty}{\|x\|_\infty} \leq c(n) \operatorname{cond}(A, x) \varepsilon\]
where
\[\operatorname{cond}(A,x) = || |A^{-1}| |A| |x|_\infty || / ||x||_\infty \leq ||A^{-1}||_\infty ||A||_\infty = \kappa_\infty(A).\]
Note that \(\operatorname{cond}(A,x)\) can be much smaller than \(\kappa_{\infty}(A)\) .
The total number of floating-point operations for one right-hand side vector is approximately \(2n^{2}\) for real flavors or \(8n^{2}\) for complex flavors.
To estimate the condition number \(\kappa_{\infty}(A)\) , call ?spcon (Estimates the reciprocal of the condition number of a packed symmetric matrix.) .
To refine the solution and estimate the error, call ?sprfs (Refines the solution of a system of linear equations with a packed symmetric coefficient matrix and estimates the solution error.) .