Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?potrf
Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix.
Syntax
call spotrf ( uplo , n , a , lda , info )
call dpotrf ( uplo , n , a , lda , info )
call cpotrf ( uplo , n , a , lda , info )
call zpotrf ( uplo , n , a , lda , info )
call potrf ( a [ , uplo ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
spotrf dpotrf cpotrf zpotrf potrf
The routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian positive-definite matrix \(A\) :
\(A = U^{T} U\) for real data, \(A = U^{H} U\) for complex data |
if uplo='U' |
\(A = L L^{T}\) for real data, \(A = L L^{H}\) for complex data |
if uplo='L' |
where \(L\) is a lower triangular matrix and \(U\) is upper triangular.
This routine supports the Progress Routine feature. See Progress Function for details.
Input Parameters
uplo
CHARACTER*1. Must be 'U' or 'L' .
Indicates whether the upper or lower triangular part of A is stored and how A is factored: If uplo = 'U' , the array a stores the upper triangular part of the matrix A , and the strictly lower triangular part of the matrix is not referenced. If uplo = 'L' , the array a stores the lower triangular part of the matrix A , and the strictly upper triangular part of the matrix is not referenced.
n
INTEGER. Specifies the order of the matrix A . The value of n must be at least zero.
- a
-
Array, size max(1, lda * n ). The array a contains either the upper or the lower triangular part of the matrix A (see uplo ). REAL for spotrfDOUBLE PRECISION for dpotrfCOMPLEX for cpotrfDOUBLE COMPLEX for zpotrf . Array, size ( lda ,*). The array a contains either the upper or the lower triangular part of the matrix A (see uplo ). The second dimension of a must be at least max(1, n) .
- lda
-
The leading dimension of a . Must be at least max(1, n ). INTEGER . The leading dimension of a .
Output Parameters
- a
-
The upper or lower triangular part of a is overwritten by the Cholesky factor U or L , as specified by uplo .
info
INTEGER .
If info=0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
If info = i , the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the factorization could not be completed. This may indicate an error in forming the matrix A .
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 potrf interface are as follows:
a
Holds the matrix A of size ( n , n ).
uplo
Must be 'U' or 'L' . The default value is 'U' .
Application Notes
If uplo='U', the computed factor \(U\) is the exact factor of a perturbed matrix \(A + E\) , where
\[|E| \leq c(n) \epsilon |U^H| |U|, \quad |e_{ij}| \leq c(n) \epsilon \sqrt{a_{ii}a_{jj}}\]
c(n) is a modest linear function of n , and \(\varepsilon\) is the machine precision.
A similar estimate holds for uplo = 'L' .
The total number of floating-point operations is approximately \((1/3)n^{3}\) for real flavors or \((4/3)n^{3}\) for complex flavors.
After calling this routine, you can call the following routines:
?potrs (Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite coefficient matrix.)
to solve \(A X = B\)
?pocon (Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite matrix.)
to estimate the condition number of \(A\)
?potri (Computes the inverse of a symmetric (Hermitian) positive-definite matrix using the Cholesky factorization.)
to compute the inverse of \(A\) .