f07jvc computes error bounds and refines the solution to a complex system of linear equations , where is an Hermitian positive definite tridiagonal matrix and and are matrices, using the modified Cholesky factorization returned by f07jrc and an initial solution returned by f07jsc. Iterative refinement is used to reduce the backward error as much as possible.
The function may be called by the names: f07jvc, nag_lapacklin_zptrfs or nag_zptrfs.
3Description
f07jvc should normally be preceded by calls to f07jrcandf07jsc. f07jrc computes a modified Cholesky factorization of the matrix as
where is a unit lower bidiagonal matrix and is a diagonal matrix, with positive diagonal elements. f07jsc then utilizes the factorization to compute a solution, , to the required equations. Letting denote a column of , f07jvc computes a component-wise backward error, , the smallest relative perturbation in each element of and such that is the exact solution of a perturbed system
The function also estimates a bound for the component-wise forward error in the computed solution defined by , where is the corresponding column of the exact solution, .
Note that the modified Cholesky factorization of can also be expressed as
where is unit upper bidiagonal.
4References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
5Arguments
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_UploTypeInput
On entry: specifies the form of the factorization as follows:
.
.
Constraint:
or .
3: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
5: – const doubleInput
Note: the dimension, dim, of the array d
must be at least
.
On entry: must contain the diagonal elements of the matrix of .
6: – const ComplexInput
Note: the dimension, dim, of the array e
must be at least
.
On entry: if , e must contain the superdiagonal elements of the matrix .
If , e must contain the subdiagonal elements of the matrix .
7: – const doubleInput
Note: the dimension, dim, of the array df
must be at least
.
On entry: must contain the diagonal elements of the diagonal matrix from the factorization of .
8: – const ComplexInput
Note: the dimension, dim, of the array ef
must be at least
.
On entry: if , ef must contain the superdiagonal elements of the unit upper bidiagonal matrix from the factorization of .
If , ef must contain the subdiagonal elements of the unit lower bidiagonal matrix from the factorization of .
9: – const ComplexInput
Note: the dimension, dim, of the array b
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: the matrix of right-hand sides .
10: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
if ,
;
if , .
11: – ComplexInput/Output
Note: the dimension, dim, of the array x
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: the initial solution matrix .
On exit: the refined solution matrix .
12: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
if ,
;
if , .
13: – doubleOutput
On exit: estimate of the forward error bound for each computed solution vector, such that , where is the th column of the computed solution returned in the array x and is the corresponding column of the exact solution . The estimate is almost always a slight overestimate of the true error.
14: – doubleOutput
On exit: estimate of the component-wise relative backward error of each computed solution vector (i.e., the smallest relative change in any element of or that makes an exact solution).
15: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
7Accuracy
The computed solution for a single right-hand side, , satisfies an equation of the form
where
and is the machine precision. An approximate error bound for the computed solution is given by
where , the condition number of with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Function f07juc can be used to compute the condition number of .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07jvc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07jvc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of floating-point operations required to solve the equations is proportional to . At most five steps of iterative refinement are performed, but usually only one or two steps are required.