NAG Library Routine Document

F07WSF (ZPFTRS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07WSF (ZPFTRS) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,

A X = B,

using the Cholesky factorization computed by F07WRF (ZPFTRF) stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

2 Specification

SUBROUTINE F07WSF (

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

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

The routine may be called by its LAPACK name zpftrs.

3 Description

F07WSF (ZPFTRS) is used to solve a complex Hermitian positive definite system of linear equations

A X = B

, the routine must be preceded by a call to F07WRF (ZPFTRF) which computes the Cholesky factorization of

A

, stored in RFP format. The solution

X

is computed by forward and backward substitution.

UPLO ='U'

A = U^{H} U

, where

U

is upper triangular; the solution

X

is computed by solving

U^{H} Y = B

and then

U X = Y

UPLO ='L'

A = L L^{H}

, where

L

is lower triangular; the solution

X

is computed by solving

L Y = B

and then

L^{H} 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 normal RFP representation of

A

or its conjugate transpose is stored.

$TRANSR ='N'$: The matrix $A$ is stored in normal RFP format.
$TRANSR ='C'$: The conjugate transpose of the RFP representation of the matrix $A$ is stored.

Constraint:

TRANSR ='N'

'C'

2: UPLO – CHARACTER(1)Input

On entry: specifies how

A

has been factorized.

$UPLO ='U'$: $A = U^{H} U$ , where $U$ is upper triangular.
$UPLO ='L'$: $A = L L^{H}$ , 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: A( $N \times (N + 1) / 2$ ) – COMPLEX (KIND=nag_wp) arrayInput

On entry: the Cholesky factorization of

A

stored in RFP format, as returned by F07WRF (ZPFTRF).

6: B(LDB, $*$ ) – COMPLEX (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 F07WSF (ZPFTRS) 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

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th parameter 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 $UPLO ='U'$ , $|E| \leq c (n) ε |U^{H}| |U|$ ;
if $UPLO ='L'$ , $|E| \leq c (n) ε |L| |L^{H}|$ ,

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 Further Comments

The total number of real floating point operations is approximately

8 n^{2} r

The real analogue of this routine is F07WEF (DPFTRS).

9 Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array})

and

B = (\begin{array}{r} 3.93 - 6.14 i & 1.48 + 6.58 i \\ 6.17 + 9.42 i & 4.65 - 4.75 i \\ - 7.17 - 21.83 i & - 4.91 + 2.29 i \\ 1.99 - 14.38 i & 7.64 - 10.79 i \end{array}) .

Here

A

is Hermitian positive definite, stored in RFP format, and must first be factorized by F07WRF (ZPFTRF).

NAG Library Routine DocumentF07WSF (ZPFTRS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07WSF (ZPFTRS)