void	nag_real_sym_posdef_band_lin_solve (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, double ab[], Integer pdab, double b[], Integer pdb, double rcond, double errbnd, NagError *fail)

3

Description

The Cholesky factorization is used to factor

A

A = U^{T} U

, if

uplo = Nag_Upper

, or

A = L L^{T}

, if

uplo = Nag_Lower

, where

U

is an upper triangular band matrix with

k

superdiagonals, and

L

is a lower triangular band matrix with

k

subdiagonals. The factored form of

A

is then used to solve the system of equations

A X = B

4

References

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 http://www.netlib.org/lapack/lug

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5

Arguments

1: $order$ – 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

order = Nag_RowMajor

. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $uplo$ – Nag_UploTypeInput

On entry: if

uplo = Nag_Upper

, the upper triangle of the matrix

A

is stored.

uplo = Nag_Lower

, the lower triangle of the matrix

A

is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $n$ – IntegerInput

On entry: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

n \geq 0

4: $kd$ – IntegerInput

On entry: the number of superdiagonals

k

(and the number of subdiagonals) of the band matrix

A

Constraint:

kd \geq 0

5: $nrhs$ – IntegerInput

On entry: the number of right-hand sides

r

, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

6: $ab [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array ab must be at least

\max (1, pdab \times n)

On entry:

if uplo=Nag_Upper then
- if $order = Nag_ColMajor$ , $a_{i j}$ is stored in $ab [(j - 1) \times pdab + kd + i - j]$ ;
- if $order = Nag_RowMajor$ , $a_{i j}$ is stored in $ab [(i - 1) \times pdab + j - i]$ ,
for max1,j-kd≤i≤j;
if uplo=Nag_Lower then
- if $order = Nag_ColMajor$ , $a_{i j}$ is stored in $ab [(j - 1) \times pdab + i - j]$ ;
- if $order = Nag_RowMajor$ , $a_{i j}$ is stored in $ab [(i - 1) \times pdab + kd + j - i]$ ,
for j≤i≤minn,j+kd,

where

pdab \geq kd + 1

is the stride separating diagonal matrix elements in the array ab.

See Section 9 below for further details.

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, the factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

, in the same storage format as

A

7: $pdab$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array ab.

Constraint:

pdab \geq kd + 1

8: $b [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times nrhs)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdb)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

B

is stored in

$b [(j - 1) \times pdb + i - 1]$ when $order = Nag_ColMajor$ ;
$b [(i - 1) \times pdb + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

r

matrix of right-hand sides

B

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, the

n

r

solution matrix

X

9: $pdb$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, nrhs)$ .

10: $rcond$ – double *Output

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, an estimate of the reciprocal of the condition number of the matrix

A

, computed as

rcond = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})

11: $errbnd$ – double *Output

On exit: if

fail . code =

NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution

\hat{x}

, such that

{‖\hat{x} - x‖}_{1} / {‖x‖}_{1} \leq errbnd

, where

\hat{x}

is a column of the computed solution returned in the array b and

x

is the corresponding column of the exact solution

X

. If rcond is less than machine precision, errbnd is returned as unity.

12: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
The Integer allocatable memory required is n, and the double allocatable memory required is $3 \times n$ . Allocation failed before the solution could be computed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $kd = 〈value〉$ .
Constraint: $kd \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $nrhs = 〈value〉$ .
Constraint: $nrhs \geq 0$ .

On entry, $pdab = 〈value〉$ .
Constraint: $pdab > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pdab = 〈value〉$ and $kd = 〈value〉$ .
Constraint: $pdab \geq kd + 1$ .

On entry, $pdb = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdb \geq \max (1, n)$ .

On entry, $pdb = 〈value〉$ and $nrhs = 〈value〉$ .
Constraint: $pdb \geq \max (1, nrhs)$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_POS_DEF: The principal minor of order $〈value〉$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.
NE_RCOND: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

7

Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. nag_real_sym_posdef_band_lin_solve (f04bfc) uses the approximation

{‖E‖}_{1} = ε {‖A‖}_{1}

to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8

Parallelism and Performance

nag_real_sym_posdef_band_lin_solve (f04bfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_real_sym_posdef_band_lin_solve (f04bfc) 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.

9

Further Comments

The band storage schemes for the array ab are identical to the storage schemesfor symmetric and Hermitian band matrices in Chapter f07. See Section 3.3.4 in the f07 Chapter Introduction for details of the storage schemes and an illustrated example.

uplo = Nag_Upper

then the elements of the stored upper triangular part of

A

are overwritten by the corresponding elements of the upper triangular matrix

U

. Similarly, if

uplo = Nag_Lower

then the elements of the stored lower triangular part of

A

are overwritten by the corresponding elements of the lower triangular matrix

L

Assuming that

n ≫ k

, the total number of floating-point operations required to solve the equations

A X = B

is approximately

{n (k + 1)}^{2}

for the factorization and

4 n k r

for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The complex analogue of nag_real_sym_posdef_band_lin_solve (f04bfc) is nag_herm_posdef_band_lin_solve (f04cfc).

10

Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite band matrix

A = (\begin{matrix} 5.49 & 2.68 & 0 & 0 \\ 2.68 & 5.63 & - 2.39 & 0 \\ 0 & - 2.39 & 2.60 & - 2.22 \\ 0 & 0 & - 2.22 & 5.17 \end{matrix}) and B = (\begin{matrix} 22.09 & 5.10 \\ 9.31 & 30.81 \\ - 5.24 & - 25.82 \\ 11.83 & 22.90 \end{matrix}) .

An estimate of the condition number of

A

and an approximate error bound for the computed solutions are also printed.

NAG Library Function Document

nag_real_sym_posdef_band_lin_solve (f04bfc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results