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NAG Toolbox: nag_lapack_dpftrs (f07we)

Purpose

nag_lapack_dpftrs (f07we) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
AX = B ,
AX=B ,
where AA has been factorized by nag_lapack_dpftrf (f07wd), stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

Syntax

[b, info] = f07we(transr, uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dpftrs(transr, uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dpftrs (f07we) is used to solve a real symmetric positive definite system of linear equations AX = BAX=B, the function must be preceded by a call to nag_lapack_dpftrf (f07wd) which computes the Cholesky factorization of AA, where AA is stored in RFP format. The solution XX is computed by forward and backward substitution.
If uplo = 'U'uplo='U', A = UTUA=UTU, where UU is upper triangular; the solution XX is computed by solving UTY = BUTY=B and then UX = YUX=Y.
If uplo = 'L'uplo='L', A = LLTA=LLT, where LL is lower triangular; the solution XX is computed by solving LY = BLY=B and then LTX = YLTX=Y.

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

Parameters

Compulsory Input Parameters

1:     transr – string (length ≥ 1)
Specifies whether the RFP representation of AA is normal or transposed.
transr = 'N'transr='N'
The matrix AA is stored in normal RFP format.
transr = 'T'transr='T'
The matrix AA is stored in transposed RFP format.
Constraint: transr = 'N'transr='N' or 'T''T'.
2:     uplo – string (length ≥ 1)
Specifies how AA has been factorized.
uplo = 'U'uplo='U'
A = UTUA=UTU, where UU is upper triangular.
uplo = 'L'uplo='L'
A = LLTA=LLT, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
3:     a(n × (n + 1) / 2n×(n+1)/2) – double array
The Cholesky factorization of AA stored in RFP format, as returned by nag_lapack_dpftrf (f07wd).
4:     b(ldb, : :) – double array
The first dimension of the array b must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,nrhs)max(1,nrhs)
The nn by rr right-hand side matrix BB.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
rr, the number of right-hand sides.
Constraint: nrhs0nrhs0.

Input Parameters Omitted from the MATLAB Interface

ldb

Output Parameters

1:     b(ldb, : :) – double array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,nrhs)max(1,nrhs)
ldbmax (1,n)ldbmax(1,n).
The nn by rr solution matrix XX.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: transr, 2: uplo, 3: n, 4: nrhs_p, 5: a, 6: b, 7: ldb, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

For each right-hand side vector bb, the computed solution xx is the exact solution of a perturbed system of equations (A + E)x = b(A+E)x=b, where c(n)c(n) is a modest linear function of nn, and εε is the machine precision
If x^ is the true solution, then the computed solution xx satisfies a forward error bound of the form
(x)/(x)c(n)cond(A,x)ε
x-x^ x c(n)cond(A,x)ε
where cond(A,x) = |A1||A||x| / xcond(A) = |A1||A|κ(A)cond(A,x)=|A-1||A||x|/xcond(A)=|A-1||A|κ(A) and κ(A)κ(A) is the condition number when using the -norm.
Note that cond(A,x)cond(A,x) can be much smaller than cond(A)cond(A).

Further Comments

The total number of floating point operations is approximately 2n2r2n2r.
The complex analogue of this function is nag_lapack_zpftrs (f07ws).

Example

function nag_lapack_dpftrs_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
b = [  8.70,  8.30;
     -13.35,  2.13;
       1.89,  1.61;
      -4.14,  5.00];
transr = 'n';
uplo   = 'l';
n      = int64(4);
% Factorize a
[aOut, info] = nag_lapack_dpftrf(transr, uplo, n, a);

if info == 0
  % Compute solution
  [bOut, info] = nag_lapack_dpftrs(transr, uplo, aOut, b);
  fprintf('\n');
  [ifail] = nag_file_print_matrix_real_gen('g', ' ', bOut, 'Solutions');
else
  fprintf('\na is not positive definite.\n');
end
 

 Solutions
             1          2
 1      1.0000     4.0000
 2     -1.0000     3.0000
 3      2.0000     2.0000
 4     -3.0000     1.0000

function f07we_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
b = [  8.70,  8.30;
     -13.35,  2.13;
       1.89,  1.61;
      -4.14,  5.00];
transr = 'n';
uplo   = 'l';
n      = int64(4);
% Factorize a
[aOut, info] = f07wd(transr, uplo, n, a);

if info == 0
  % Compute solution
  [bOut, info] = f07we(transr, uplo, aOut, b);
  fprintf('\n');
  [ifail] = x04ca('g', ' ', bOut, 'Solutions');
else
  fprintf('\na is not positive definite.\n');
end
 

 Solutions
             1          2
 1      1.0000     4.0000
 2     -1.0000     3.0000
 3      2.0000     2.0000
 4     -3.0000     1.0000


PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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