Integer, Intent (In)	::	n, kd, nrhs, ldab, ldb
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Inout)	::	ab(ldab,), b(ldb,)
Real (Kind=nag_wp), Intent (Out)	::	rcond, errbnd
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f04bff_ (const char uplo, const Integer n, const Integer kd, const Integer nrhs, double ab[], const Integer ldab, double b[], const Integer ldb, double rcond, double errbnd, Integer *ifail, const Charlen length_uplo)

The routine may be called by the names f04bff or nagf_linsys_real_posdef_band_solve.

3 Description

The Cholesky factorization is used to factor

A

A = U^{T} U

, if

uplo ='U'

, or

A = L L^{T}

, if

uplo ='L'

, 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 https://www.netlib.org/lapack/lug

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

5 Arguments

1: $uplo$ – Character(1) Input

On entry: if

uplo ='U'

, the upper triangle of the matrix

A

is stored.

uplo ='L'

, the lower triangle of the matrix

A

is stored.

Constraint:

uplo ='U'

'L'

2: $n$ – Integer Input

On entry: the number of linear equations

n

, i.e., the order of the matrix

A

Constraint:

n \geq 0

3: $kd$ – Integer Input

On entry: the number of superdiagonals

k

(and the number of subdiagonals) of the band matrix

A

Constraint:

kd \geq 0

4: $nrhs$ – Integer Input

On entry: the number of right-hand sides

r

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

B

Constraint:

nrhs \geq 0

5: $ab (ldab, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array ab must be at least

\max (1, n)

On entry: the

n \times n

symmetric band matrix

A

. The upper or lower triangular part of the symmetric matrix is stored in the first

kd + 1

rows of the array. The

j

th column of

A

is stored in the

j

th column of the array ab as follows:

The matrix is stored in rows

1

k + 1

, more precisely,

if $uplo ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (k + 1 + i - j, j) for \max (1, j - k) \leq i \leq j$ ;
if $uplo ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (1 + i - j, j) for j \leq i \leq \min (n, j + k) .$

See Section 9 below for further details.

On exit: if

ifail = 0

n + 1

, the factor

U

L

from the Cholesky factorization

A = U^{T} U

A = L L^{T}

, in the same storage format as

A

6: $ldab$ – Integer Input

On entry: the first dimension of the array ab as declared in the (sub)program from which f04bff is called.

Constraint:

ldab \geq kd + 1

7: $b (ldb, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array b must be at least

\max (1, nrhs)

On entry: the

n \times r

matrix of right-hand sides

B

On exit: if

ifail = 0

n + 1

, the

n \times r

solution matrix

X

8: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f04bff is called.

Constraint:

ldb \geq \max (1, n)

9: $rcond$ – Real (Kind=nag_wp) Output

On exit: if

ifail = 0

n + 1

, an estimate of the reciprocal of the condition number of the matrix

A

, computed as

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

10: $errbnd$ – Real (Kind=nag_wp) Output

On exit: if

ifail = 0

n + 1

, 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.

11: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail > 0 and ifail \leq n$: 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.

$ifail = n + 1$: A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.

$ifail = - 1$: On entry, uplo not one of 'U' or 'u' or 'L' or 'l': $uplo = ⟨ value ⟩$ .

$ifail = - 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

$ifail = - 3$: On entry, $kd = ⟨ value ⟩$ .
Constraint: $kd \geq 0$ .

$ifail = - 4$: On entry, $nrhs = ⟨ value ⟩$ .
Constraint: $nrhs \geq 0$ .

$ifail = - 6$: On entry, $ldab = ⟨ value ⟩$ and $kd = ⟨ value ⟩$ .
Constraint: $ldab \geq kd + 1$ .

$ifail = - 8$: On entry, $ldb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldb \geq \max (1, n)$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
The integer allocatable memory required is n, and the real allocatable memory required is $3 \times n$ . Allocation failed before the solution could be computed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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. f04bff 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

f04bff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f04bff 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The band storage scheme for the array ab is illustrated by the following example, when

n = 6

k = 2

, and

uplo ='U'

On entry:

\begin{matrix} * & * & a_{13} & a_{24} & a_{35} & a_{46} \\ * & a_{12} & a_{23} & a_{34} & a_{45} & a_{56} \\ a_{11} & a_{22} & a_{33} & a_{44} & a_{55} & a_{66} \end{matrix}

On exit:

\begin{matrix} * & * & u_{13} & u_{24} & u_{35} & u_{46} \\ * & u_{12} & u_{23} & u_{34} & u_{45} & u_{56} \\ u_{11} & u_{22} & u_{33} & u_{44} & u_{55} & u_{66} \end{matrix}

Similarly, if

uplo ='L'

the format of ab is as follows:

On entry:

\begin{matrix} a_{11} & a_{22} & a_{33} & a_{44} & a_{55} & a_{66} \\ a_{21} & a_{32} & a_{43} & a_{54} & a_{65} & * \\ a_{31} & a_{42} & a_{53} & a_{64} & * & * \end{matrix}

On exit:

\begin{matrix} l_{11} & l_{22} & l_{33} & l_{44} & l_{55} & l_{66} \\ l_{21} & l_{32} & l_{43} & l_{54} & l_{65} & * \\ l_{31} & l_{42} & l_{53} & l_{64} & * & * \end{matrix}

Array elements marked

*

need not be set and are not referenced by the routine.

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 f04bff is f04cff.

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.

f04bf: FL CL CPP AD

NAG FL Interfacef04bff (real_​posdef_​band_​solve)

▸▿ 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

NAG FL Interface
f04bff (real_posdef_band_solve)