Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?pbtrs
Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite band coefficient matrix.
Syntax
call spbtrs ( uplo , n , kd , nrhs , ab , ldab , b , ldb , info )
call dpbtrs ( uplo , n , kd , nrhs , ab , ldab , b , ldb , info )
call cpbtrs ( uplo , n , kd , nrhs , ab , ldab , b , ldb , info )
call zpbtrs ( uplo , n , kd , nrhs , ab , ldab , b , ldb , info )
call pbtrs ( ab , b [ , uplo ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
spbtrs dpbtrs cpbtrs zpbtrs pbtrs
The routine solves for real data a system of linear equations A*X = B with a symmetric positive-definite or, for complex data, Hermitian positive-definite band matrix A , given the Cholesky factorization of 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. The system is solved with multiple right-hand sides stored in the columns of the matrix B .
Before calling this routine, you must call ?pbtrf (Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite band matrix.) to compute the Cholesky factorization of A in the band storage form.
Input Parameters
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ .
Indicates how the input matrix A has been factored:
If uplo = 'U' , U is stored in ab , where A = UT * U for real matrices and A = UH * U for complex matrices.
If uplo = 'L' , L is stored in ab , where A = L * LT for real matrices and A = L * LH for complex matrices.
n
INTEGER . The order of matrix A ; n ≥ 0.
kd
INTEGER . The number of superdiagonals or subdiagonals in the matrix A ; kd ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
ab , b
REAL for spbtrs
DOUBLE PRECISION for dpbtrs
COMPLEX for cpbtrs
DOUBLE COMPLEX for zpbtrs .
Arrays: ab(ldab,*) , b(ldb,*) .
The array ab contains the Cholesky factor, as returned by the factorization routine, in band storage form.
The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.
The second dimension of ab must be at least max(1, n) , and the second dimension of b at least max(1,nrhs) .
b
The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.
The size of b is at least max(1, ldb * nrhs ) for column major layout and max(1, ldb * n ) for row major layout.
ldab
INTEGER . The leading dimension of the array ab ; ldab ≥ kd +1.
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 pbtrs interface are as follows:
ab
Holds the array A of size (kd+1,n) .
b
Holds the matrix B of size ( n , nrhs ).
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
\[|E| \leq c(kd+1) \varepsilon P|U||D||U^{H}|P^{T} or |E| \le c(n) \varepsilon P|L||D||L^{H}|P^{T}\]
\(c(n)\) 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:
\[\frac{\|x - x_0\|_\infty}{\|x\|_\infty} \leq c(kd+1) \operatorname{cond}(A, x) \varepsilon\]
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 approximate number of floating-point operations for one right-hand side vector is \(4*n*kd\) for real flavors and \(16*n*kd\) for complex flavors.
To estimate the condition number \(\kappa_{\infty}(A)\) , call ?pbcon (Estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite band matrix.) .
To refine the solution and estimate the error, call ?pbrfs (Refines the solution of a system of linear equations with a band symmetric (Hermitian) positive-definite coefficient matrix and estimates its error.) .