Developer Reference for Intel® oneAPI Math Kernel Library for C
?pbsvx
Uses the Cholesky factorization to compute the solution to the system of linear equations with a symmetric (Hermitian) positive-definite band coefficient matrix A, and provides error bounds on the solution.
Syntax
lapack_int LAPACKE_spbsvx ( intmatrix_layout , charfact , charuplo , lapack_intn , lapack_intkd , lapack_intnrhs , float*ab , lapack_intldab , float*afb , lapack_intldafb , char*equed , float*s , float*b , lapack_intldb , float*x , lapack_intldx , float*rcond , float*ferr , float*berr );
lapack_int LAPACKE_dpbsvx ( intmatrix_layout , charfact , charuplo , lapack_intn , lapack_intkd , lapack_intnrhs , double*ab , lapack_intldab , double*afb , lapack_intldafb , char*equed , double*s , double*b , lapack_intldb , double*x , lapack_intldx , double*rcond , double*ferr , double*berr );
lapack_int LAPACKE_cpbsvx ( intmatrix_layout , charfact , charuplo , lapack_intn , lapack_intkd , lapack_intnrhs , lapack_complex_float*ab , lapack_intldab , lapack_complex_float*afb , lapack_intldafb , char*equed , float*s , lapack_complex_float*b , lapack_intldb , lapack_complex_float*x , lapack_intldx , float*rcond , float*ferr , float*berr );
lapack_int LAPACKE_zpbsvx ( intmatrix_layout , charfact , charuplo , lapack_intn , lapack_intkd , lapack_intnrhs , lapack_complex_double*ab , lapack_intldab , lapack_complex_double*afb , lapack_intldafb , char*equed , double*s , lapack_complex_double*b , lapack_intldb , lapack_complex_double*x , lapack_intldx , double*rcond , double*ferr , double*berr );
Include Files
mkl.h
Description
spbsvx dpbsvx cpbsvx zpbsvx pbsvx
The routine uses the Cholesky factorization A=U^{T}*U (real flavors) / A=U^{H}*U (complex flavors) or A=L*L^{T} (real flavors) / A=L*L^{H} (complex flavors) to compute the solution to a real or complex system of linear equations A*X = B , where A is a n -by- n symmetric or Hermitian positive definite band matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.
Error bounds on the solution and a condition estimate are also provided.
The routine ?pbsvx performs the following steps:
If fact = 'E' , real scaling factors s are computed to equilibrate the system:
diag(s)* A * diag(s)* inv(diag(s)) * X = diag(s)* B .
Whether or not the system will be equilibrated depends on the scaling of the matrix A , but if equilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B .
If fact = 'N' or ‘E’ , the Cholesky decomposition is used to factor the matrix A (after equilibration if fact = 'E' ) as
A = U^{T}*U (real), A = U^{H}*U (complex), if uplo = 'U' ,
or A = L*L^{T} (real), A = L*L^{H} (complex), if uplo = 'L' ,
where U is an upper triangular band matrix and L is a lower triangular band matrix.
If the leading i -by- i principal minor is not positive definite, then the routine returns with info = i . Otherwise, the factored form of A is used to estimate the condition number of the matrix A . If the reciprocal of the condition number is less than machine precision, info = n+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
The system of equations is solved for X using the factored form of A .
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.
If equilibration was used, the matrix X is premultiplied by diag(s) so that it solves the original system before equilibration.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major ( LAPACK_ROW_MAJOR ) or column major ( LAPACK_COL_MAJOR ).
fact
Must be ‘F’ , ‘N’ , or ‘E’ .
Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.
If fact = 'F' : on entry, afb contains the factored form of A . If equed = 'Y' , the matrix A has been equilibrated with scaling factors given by s . ab and afb will not be modified.
If fact = 'N' , the matrix A will be copied to afb and factored.
If fact = 'E' , the matrix A will be equilibrated if necessary, then copied to afb and factored.
uplo
Must be ‘U’ or ‘L’ .
Indicates whether the upper or lower triangular part of A is stored:
If uplo = 'U' , the upper triangle of A is stored.
If uplo = 'L' , the lower triangle of A is stored.
n
The order of matrix A ; n ≥ 0.
kd
The number of superdiagonals or subdiagonals in the matrix A ; kd ≥ 0.
nrhs
The number of right-hand sides, the number of columns in B; nrhs≥ 0 .
ab , afb , b , work
Arrays: ab (size max(1, ldab * n )) , afb (size max(1, ldafb * n )) , b of size max(1, ldb * nrhs ) for column major layout and max(1, ldb * n ) for row major layout .
The array ab contains the upper or lower triangle of the matrix A in band storage (see Matrix Storage Schemes ).
If fact = 'F' and equed = ‘Y’ , then ab must contain the equilibrated matrix diag(s)*A*diag(s) .
The array afb is an input argument if fact = 'F' . It contains the triangular factor U or L from the Cholesky factorization of the band matrix A in the same storage format as A . If equed = 'Y' , then afb is the factored form of the equilibrated matrix A .
The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.
ldab
The leading dimension of ab ; ldab ≥ kd +1.
ldafb
The leading dimension of afb ; ldafb ≥ kd +1.
ldb
The leading dimension of b ; ldb≥ max(1, n)ldb ≥ max(1, n ) for column major layout and ldb ≥ nrhs for row major layout .
equed
Must be ‘N’ or ‘Y’ .
equed is an input argument if fact = 'F' . It specifies the form of equilibration that was done:
if equed = 'N' , no equilibration was done (always true if fact = 'N' )
if equed = 'Y' , equilibration was done, that is, A has been replaced by diag(s)*A*diag(s) .
s
Array, size ( n ). The array s contains the scale factors for A . This array is an input argument if fact = 'F' only; otherwise it is an output argument.
If equed = 'N' , s is not accessed.
If fact = 'F' and equed = ‘Y’ , each element of s must be positive.
ldx
The leading dimension of the output array x ; ldx≥ max(1, n)ldx ≥ max(1, n ) for column major layout and ldx ≥ nrhs for row major layout .
iwork
Workspace array, size at least max(1, n) ; used in real flavors only.
rwork
Workspace array, size at least max(1, n) ; used in complex flavors only.
Output Parameters
x
Array, size max(1, ldx * nrhs ) for column major layout and max(1, ldx * n ) for row major layout .
If info = 0 or info = n+1 , the array x contains the solution matrix X to the original system of equations. Note that if equed = 'Y' , A and B are modified on exit, and the solution to the equilibrated system is inv(diag(s))*X .
ab
On exit, if fact = 'E' and equed = ‘Y’ , A is overwritten by diag(s)*A*diag(s).
afb
If fact = 'N' or ‘E’ , then afb is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A=U^{T}*U or A=L*L^{T} (real routines), A=U^{H}*U or A=L*L^{H} (complex routines) of the original matrix A (if fact = 'N' ), or of the equilibrated matrix A (if fact = 'E' ). See the description of ab for the form of the equilibrated matrix.
b
Overwritten by diag(s)*B , if equed = 'Y' ; not changed if equed = ‘N’ .
s
This array is an output argument if fact≠'F' . See the description of s in Input Arguments section.
rcond
An estimate of the reciprocal condition number of the matrix A after equilibration (if done). If rcond is less than the machine precision (in particular, if rcond = 0), the matrix is singular to working precision. This condition is indicated by a return code of info > 0.
ferr
Array, size at least max(1, nrhs) . Contains the estimated forward error bound for each solution vector x(j)x_{j} (the j -th column of the solution matrix X ). If xtrue is the true solution corresponding to x(j)x_{j} , ferr(j)ferr[j-1] is an estimated upper bound for the magnitude of the largest element in (x(j) - xtrue)(x_{j} - xtrue) divided by the magnitude of the largest element in x(j)x_{j} . The estimate is as reliable as the estimate for rcond , and is almost always a slight overestimate of the true error.
berr
Array, size at least max(1, nrhs) . Contains the component-wise relative backward error for each solution vector x(j)x_{j} , that is, the smallest relative change in any element of A or B that makes x(j)x_{j} an exact solution.
equed
If fact≠'F' , then equed is an output argument. It specifies the form of equilibration that was done (see the description of equed in Input Arguments section).
Return Values
This function returns a value info .
If info = 0 , the execution is successful.
If info = -i , parameter i had an illegal value.
If info = i , and i≤n , the leading minor of order i (and therefore the matrix A itself) is not positive definite, so the factorization could not be completed, and the solution and error bounds could not be computed; rcond =0 is returned. If info = i , and i = n + 1, then U is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.
LAPACK 95 Interface Notes
There exist FORTRAN 77 and FORTRAN 95 interfaces for this routine. See the Intel® oneMKL Fortran Developer Reference for details.