NAG Library Routine Document

F07WEF (DPFTRS)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

F07WEF (DPFTRS) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,

A X = B,

using the Cholesky factorization computed by F07WDF (DPFTRF) stored in Rectangular Full Packed (RFP) format.

2 Specification

SUBROUTINE F07WEF (

TRANSR, UPLO, N, NRHS, AR, B, LDB, INFO)

INTEGER	N, NRHS, LDB, INFO
REAL (KIND=nag_wp)	AR(N(N+1)/2), B(LDB,)
CHARACTER(1)	TRANSR, UPLO

The routine may be called by its LAPACK name dpftrs.

3 Description

F07WEF (DPFTRS) is used to solve a real symmetric positive definite system of linear equations

A X = B

, the routine must be preceded by a call to F07WDF (DPFTRF) which computes the Cholesky factorization of

A

, stored in RFP format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction. The solution

X

is computed by forward and backward substitution.

UPLO ='U'

A = U^{T} U

, where

U

is upper triangular; the solution

X

is computed by solving

U^{T} Y = B

and then

U X = Y

UPLO ='L'

A = L L^{T}

, where

L

is lower triangular; the solution

X

is computed by solving

L Y = B

and then

L^{T} X = Y

4 References

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

5 Parameters

1: $TRANSR$ – CHARACTER(1)Input

On entry: specifies whether the RFP representation of

A

is normal or transposed.

$TRANSR ='N'$: The matrix $A$ is stored in normal RFP format.
$TRANSR ='T'$: The matrix $A$ is stored in transposed RFP format.

Constraint:

TRANSR ='N'

'T'

2: $UPLO$ – CHARACTER(1)Input

On entry: specifies how

A

has been factorized.

$UPLO ='U'$: $A = U^{T} U$ , where $U$ is upper triangular.
$UPLO ='L'$: $A = L L^{T}$ , where $L$ is lower triangular.

Constraint:

UPLO ='U'

'L'

3: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

4: $NRHS$ – INTEGERInput

On entry:

r

, the number of right-hand sides.

Constraint:

NRHS \geq 0

5: $AR (N \times (N + 1) / 2)$ – REAL (KIND=nag_wp) arrayInput

On entry: the Cholesky factorization of

A

stored in RFP format, as returned by F07WDF (DPFTRF).

6: $B (LDB *)$ – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, NRHS)

On entry: the

n

r

right-hand side matrix

B

On exit: the

n

r

solution matrix

X

7: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07WEF (DPFTRS) is called.

Constraint:

LDB \geq \max (1, N)

8: $INFO$ – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$INFO < 0$: If $INFO = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

if $UPLO ='U'$ , $|E| \leq c (n) ε |U^{T}| |U|$ ;
if $UPLO ='L'$ , $|E| \leq c (n) ε |L| |L^{T}|$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (n) cond (A, x) ε

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

and

κ_{\infty} (A)

is the condition number when using the

\infty

-norm.

Note that

cond (A, x)

can be much smaller than

cond (A)

8 Parallelism and Performance

F07WEF (DPFTRS) is not threaded by NAG in any implementation.

F07WEF (DPFTRS) 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 total number of floating-point operations is approximately

2 n^{2} r

The complex analogue of this routine is F07WSF (ZPFTRS).

10 Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) and B = (\begin{matrix} 8.70 & 8.30 \\ - 13.35 & 2.13 \\ 1.89 & 1.61 \\ - 4.14 & 5.00 \end{matrix}) .

Here

A

is symmetric positive definite, stored in RFP format, and must first be factorized by F07WDF (DPFTRF).

NAG Library Routine DocumentF07WEF (DPFTRS)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

F07WEF (DPFTRS)