nag_zpotrs (f07fsc) nag_zpotrs (f07fsc) solves a complex Hermitian positive-definite system of linear equations with multiple right-hand sides, AX  B, where A has been factorized by nag_zpotrf (f07frc)

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NAG C Library Function Document
nag_zpotrs (f07fsc)
1 Purpose
nag_zpotrs (f07fsc) solves a complex Hermitian positive-deﬁnite system of linear equations with multiple
right-hand sides, AX ¼ B, where A has been factorized by nag_zpotrf (f07frc).
2 Speciﬁcation
void nag_zpotrs (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs,
const Complex a[], Integer pda, Complex b[], Integer pdb, NagError *fail)
3 Description
To solve a complex Hermitian positive-deﬁnite system of linear equations AX ¼ B, this function must be
preceded by a call to nag_zpotrf (f07frc) which computes the Cholesky factorization of A. The solution X
is computed by forward and backward substitution.
If uplo ¼ Nag Upper, A ¼ U
HU, where U is upper triangular; the solution X is computed by solving
U
HY ¼ B and then UX ¼ Y.
If uplo ¼ Nag Lower, A ¼ LL
H, where L is lower triangular; the solution X is computed by solving
LY ¼ B and then L
HX ¼ Y.
4 References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press,
Baltimore
5 Parameters
1: order – Nag_OrderType Input
On entry: the order parameter 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 ¼ Nag RowMajor. See Section 2.2.1.4 of the Essential Introduction for a more detailed
explanation of the use of this parameter.
Constraint: order ¼ Nag RowMajor or Nag ColMajor.
2: uplo – Nag_UploType Input
On entry: indicates whether A has been factorized as U
HU or LL
Has follows:
if uplo ¼ Nag Upper, then A ¼ U
HU, where U is upper triangular;
if uplo ¼ Nag Lower, then A ¼ LL
H, where L is lower triangular.
Constraint: uplo ¼ Nag Upper or Nag Lower.
3: n – Integer Input
On entry: n, the order of the matrix A.
Constraint: n   0.
f07 – Linear Equations (LAPACK) f07fsc
[NP3645/7] f07fsc.14: nrhs – Integer Input
On entry: r, the number of right-hand sides.
Constraint: nrhs   0.
5: a½dim  – const Complex Input
Note: the dimension, dim, of the array a must be at least maxð1;pda   nÞ.
On entry: the Cholesky factor of A, as returned by nag_zpotrf (f07frc).
6: pda – Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the
matrix in the array a.
Constraint: pda   maxð1;nÞ.
7: b½dim  – Complex Input/Output
Note: the dimension, dim,o ft h ea r r a yb must be at least maxð1;pdb   nrhsÞ when
order ¼ Nag ColMajor and at least maxð1;pdb   nÞ when order ¼ Nag RowMajor.
If order ¼ Nag ColMajor, the ði;jÞth element of the matrix B is stored in b½ðj   1Þ pdb þ i   1  and
if order ¼ Nag RowMajor, the ði;jÞth element of the matrix B is stored in b½ði   1Þ pdb þ j   1 .
On entry: the n by r right-hand side matrix B.
On exit: the n by r solution matrix X.
8: pdb – Integer Input
On entry: the stride separating matrix row or column elements (depending on the value of order)i n
the array b.
Constraints:
if order ¼ Nag ColMajor, pdb   maxð1;nÞ;
if order ¼ Nag RowMajor, pdb   maxð1;nrhsÞ.
9: fail – NagError * Output
The NAG error parameter (see the Essential Introduction).
6 Error Indicators and Warnings
NE_INT
On entry, n = hvaluei.
Constraint: n   0.
On entry, nrhs = hvaluei.
Constraint: nrhs   0.
On entry, pda ¼h valuei.
Constraint: pda > 0.
On entry, pdb ¼h valuei.
Constraint: pdb > 0.
NE_INT_2
On entry, pda ¼h valuei, n ¼h valuei.
Constraint: pda   maxð1;nÞ.
On entry, pdb ¼h valuei, n ¼h valuei.
Constraint: pdb   maxð1;nÞ.
f07fsc NAG C Library Manual
f07fsc.2 [NP3645/7]On entry, pdb ¼h valuei, nrhs ¼h valuei.
Constraint: pdb   maxð1;nrhsÞ.
NE_ALLOC_FAIL
Memory allocation failed.
NE_BAD_PARAM
On entry, parameter hvaluei had an illegal value.
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 consult NAG for assistance.
7 Accuracy
For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of
equations ðA þ EÞx ¼ b, where
if uplo ¼ Nag Upper, jEj cðnÞ jU
HjjUj;
if uplo ¼ Nag Lower, jEj cðnÞ jLjjL
Hj,
cðnÞ is a modest linear function of n, and   is the machine precision.
If ^ x x is the true solution, then the computed solution x satisﬁes a forward error bound of the form
kx   ^ x xk1
kxk1
  cðnÞcondðA;xÞ 
where condðA;xÞ¼k j A
 1jjAjjxjk1=kxk1   condðAÞ¼k j A
 1jjAjk1    1ðAÞ. Note that condðA;xÞ
can be much smaller than condðAÞ.
Forward and backward error bounds can be computed by calling nag_zporfs (f07fvc), and an estimate for
 1ðAÞ (¼  1ðAÞ) can be obtained by calling nag_zpocon (f07fuc).
8 Further Comments
The total number of real ﬂoating-point operations is approximately 8n
2r.
This function may be followed by a call to nag_zporfs (f07fvc) to reﬁne the solution and return an error
estimate.
The real analogue of this function is nag_dpotrs (f07fec).
9 Example
To solve the system of equations AX ¼ B, where
A ¼
3:23 þ 0:00i 1:51   1:92i 1:90 þ 0:84i 0:42 þ 2:50i
1:51 þ 1:92i 3:58 þ 0:00i  0:23 þ 1:11i  1:18 þ 1:37i
1:90   0:84i  0:23   1:11i 4:09 þ 0:00i 2:33   0:14i
0:42   2:50i  1:18   1:37i 2:33 þ 0:14i 4:29 þ 0:00i
1
C C A
0
B B @
and
B ¼
3:93   6:14i 1:48 þ 6:58i
6:17 þ 9:42i 4:65   4:75i
 7:17   21:83i  4:91 þ 2:29i
1:99   14:38i 7:64   10:79i
1
C C A
0
B B @ :
Here A is Hermitian positive-deﬁnite and must ﬁrst be factorized by nag_zpotrf (f07frc).
f07 – Linear Equations (LAPACK) f07fsc
[NP3645/7] f07fsc.39.1 Program Text
/* nag_zpotrs (f07fsc) Example Program.
*
* Copyright 2001 Numerical Algorithms Group.
*
* Mark 7, 2001.
*/
#include <stdio.h>
#include <nag.h>
#include <nag_stdlib.h>
#include <nagf07.h>
#include <nagx04.h>
int main(void)
{
/* Scalars */
Integer i, j, n, nrhs, pda, pdb;
Integer exit_status=0;
Nag_UploType uplo_enum;
NagError fail;
Nag_OrderType order;
/* Arrays */
char uplo[2];
Complex *a=0, *b=0;
#ifdef NAG_COLUMN_MAJOR
#define A(I,J) a[(J-1)*p d a+I-1 ]
#define B(I,J) b[(J-1)*p d b+I-1 ]
order = Nag_ColMajor;
#else
#define A(I,J) a[(I-1)*p d a+J-1 ]
#define B(I,J) b[(I-1)*p d b+J-1 ]
order = Nag_RowMajor;
#endif
INIT_FAIL(fail);
Vprintf("f07fsc Example Program Results\n\n");
/* Skip heading in data file */
Vscanf("%*[^\n] ");
Vscanf("%ld%ld%*[^\n] ", &n, &nrhs);
#ifdef NAG_COLUMN_MAJOR
pda = n;
pdb = n;
#else
pda = n;
pdb = nrhs;
#endif
/* Allocate memory */
if ( !(a = NAG_ALLOC(n * n, Complex)) ||
!(b = NAG_ALLOC(n * nrhs, Complex)) )
{
Vprintf("Allocation failure\n");
exit_status = -1;
goto END;
}
/* Read A and B from data file */
Vscanf(" ’ %1s ’%*[^\n] ", uplo);
if (*(unsigned char *)uplo == ’L’)
uplo_enum = Nag_Lower;
else if (*(unsigned char *)uplo == ’U’)
uplo_enum = Nag_Upper;
else
{
Vprintf("Unrecognised character for Nag_UploType type\n");
f07fsc NAG C Library Manual
f07fsc.4 [NP3645/7]exit_status = -1;
goto END;
}
if (uplo_enum == Nag_Upper)
{
for (i = 1; i <= n; ++i)
{
for (j = i; j <= n; ++j)
Vscanf(" ( %lf , %lf )", &A(i,j).re, &A(i,j).im);
}
Vscanf("%*[^\n] ");
}
else
{
for (i = 1; i <= n; ++i)
{
for (j = 1; j <= i; ++j)
Vscanf(" ( %lf , %lf )", &A(i,j).re, &A(i,j).im);
}
Vscanf("%*[^\n] ");
}
for (i = 1; i <= n; ++i)
{
for (j = 1; j <= nrhs; ++j)
Vscanf(" ( %lf , %lf )", &B(i,j).re, &B(i,j).im);
}
Vscanf("%*[^\n] ");
/* Factorize A */
f07frc(order, uplo_enum, n, a, pda, &fail);
if (fail.code != NE_NOERROR)
{
Vprintf("Error from f07frc.\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Compute solution */
f07fsc(order, uplo_enum, n, nrhs, a, pda, b, pdb, &fail);
if (fail.code != NE_NOERROR)
{
Vprintf("Error from f07fsc.\n%s\n", fail.message);
exit_status = 1;
goto END;
}
/* Print solution */
x04dbc(order, Nag_GeneralMatrix, Nag_NonUnitDiag, n, nrhs, b, pdb,
Nag_BracketForm, "%7.4f", "Solution(s)", Nag_IntegerLabels, 0,
Nag_IntegerLabels, 0, 80, 0, 0, &fail);
if (fail.code != NE_NOERROR)
{
Vprintf("Error from x04dbc.\n%s\n", fail.message);
exit_status = 1;
goto END;
}
END:
if (a) NAG_FREE(a);
if (b) NAG_FREE(b);
return exit_status;
}
9.2 Program Data
f07fsc Example Program Data
42 :Values of N and NRHS
’L’ :Value of UPLO
(3.23, 0.00)
(1.51, 1.92) ( 3.58, 0.00)
f07 – Linear Equations (LAPACK) f07fsc
[NP3645/7] f07fsc.5(1.90,-0.84) (-0.23,-1.11) ( 4.09, 0.00)
(0.42,-2.50) (-1.18,-1.37) ( 2.33, 0.14) ( 4.29, 0.00) :End of matrix A
( 3.93, -6.14) ( 1.48, 6.58)
( 6.17, 9.42) ( 4.65, -4.75)
(-7.17,-21.83) (-4.91, 2.29)
( 1.99,-14.38) ( 7.64,-10.79) :End of matrix B
9.3 Program Results
f07fsc Example Program Results
Solution(s)
12
1 ( 1.0000,-1.0000) (-1.0000, 2.0000)
2 (-0.0000, 3.0000) ( 3.0000,-4.0000)
3 (-4.0000,-5.0000) (-2.0000, 3.0000)
4 ( 2.0000, 1.0000) ( 4.0000,-5.0000)
f07fsc NAG C Library Manual
f07fsc.6 (last) [NP3645/7]

f07 – Linear Equations (LAPACK) f07fsc

http://www.nag.com/numeric/CL/nagdoc_cl08/pdf/F07/f07fsc.pdf

f07 – Linear Equations (LAPACK) f07fsc

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