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NAG Toolbox: nag_lapack_zgbtrs (f07bs)

Purpose

nag_lapack_zgbtrs (f07bs) solves a complex band system of linear equations with multiple right-hand sides,
AX = B ,  ATX = B   or   AHX = B ,
AX=B ,  ATX=B   or   AHX=B ,
where AA has been factorized by nag_lapack_zgbtrf (f07br).

Syntax

[b, info] = f07bs(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zgbtrs(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgbtrs (f07bs) is used to solve a complex band system of linear equations AX = BAX=B, ATX = BATX=B or AHX = BAHX=B, the function must be preceded by a call to nag_lapack_zgbtrf (f07br) which computes the LULU factorization of AA as A = PLUA=PLU. The solution is computed by forward and backward substitution.
If trans = 'N'trans='N', the solution is computed by solving PLY = BPLY=B and then UX = YUX=Y.
If trans = 'T'trans='T', the solution is computed by solving UTY = BUTY=B and then LTPTX = YLTPTX=Y.
If trans = 'C'trans='C', the solution is computed by solving UHY = BUHY=B and then LHPTX = YLHPTX=Y.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans = 'N'trans='N'
AX = BAX=B is solved for XX.
trans = 'T'trans='T'
ATX = BATX=B is solved for XX.
trans = 'C'trans='C'
AHX = BAHX=B is solved for XX.
Constraint: trans = 'N'trans='N', 'T''T' or 'C''C'.
2:     kl – int64int32nag_int scalar
klkl, the number of subdiagonals within the band of the matrix AA.
Constraint: kl0kl0.
3:     ku – int64int32nag_int scalar
kuku, the number of superdiagonals within the band of the matrix AA.
Constraint: ku0ku0.
4:     ab(ldab, : :) – complex array
The first dimension of the array ab must be at least 2 × kl + ku + 12×kl+ku+1
The second dimension of the array must be at least max (1,n)max(1,n)
The LULU factorization of AA, as returned by nag_lapack_zgbtrf (f07br).
5:     ipiv( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)max(1,n).
The pivot indices, as returned by nag_lapack_zgbtrf (f07br).
6:     b(ldb, : :) – complex 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 second dimension of the array ab.
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

ldab ldb

Output Parameters

1:     b(ldb, : :) – complex 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: trans, 2: n, 3: kl, 4: ku, 5: nrhs_p, 6: ab, 7: ldab, 8: ipiv, 9: b, 10: ldb, 11: 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
|E|c(k)ε|L||U| ,
|E|c(k)ε|L||U| ,
c(k)c(k) is a modest linear function of k = kl + ku + 1k=kl+ku+1, and εε is the machine precision. This assumes knkn.
If x^ is the true solution, then the computed solution xx satisfies a forward error bound of the form
(x)/(x)c(k)cond(A,x)ε
x-x^ x c(k)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).
Note that cond(A,x)cond(A,x) can be much smaller than cond(A)cond(A), and cond(AH)cond(AH) (which is the same as cond(AT)cond(AT)) can be much larger (or smaller) than cond(A)cond(A).
Forward and backward error bounds can be computed by calling nag_lapack_zgbrfs (f07bv), and an estimate for κ(A)κ(A) can be obtained by calling nag_lapack_zgbcon (f07bu) with norm = 'I'norm='I'.

Further Comments

The total number of real floating point operations is approximately 8n(2kl + ku)r8n(2kl+ku)r, assuming nklnkl and nkunku.
This function may be followed by a call to nag_lapack_zgbrfs (f07bv) to refine the solution and return an error estimate.
The real analogue of this function is nag_lapack_dgbtrs (f07be).

Example

function nag_lapack_zgbtrs_example
trans = 'N';
m = int64(4);
kl = int64(1);
ku = int64(2);
ab = [complex(0),  0 + 0i,  0 + 0i,  0 + 0i;
      0 + 0i,  0 + 0i,  0.97 - 2.84i,  0.59 - 0.48i;
      0 + 0i,  -2.05 - 0.85i,  -3.99 + 4.01i,  3.33 - 1.04i;
      -1.65 + 2.26i,  -1.48 - 1.75i,  -1.06 + 1.94i,  -0.46 - 1.72i;
      0 + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0 + 0i];
b = [ -1.06 + 21.5i,  12.85 + 2.84i;
      -22.72 - 53.9i,  -70.22 + 21.57i;
      28.24 - 38.6i,  -20.73 - 1.23i;
      -34.56 + 16.73i,  26.01 + 31.97i];
[ab, ipiv, info] = nag_lapack_zgbtrf(m, kl, ku, ab);
[bOut, info] = nag_lapack_zgbtrs(trans, kl, ku, ab, ipiv, b)
 

bOut =

  -3.0000 + 2.0000i   1.0000 + 6.0000i
   1.0000 - 7.0000i  -7.0000 - 4.0000i
  -5.0000 + 4.0000i   3.0000 + 5.0000i
   6.0000 - 8.0000i  -8.0000 + 2.0000i


info =

                    0


function f07bs_example
trans = 'N';
m = int64(4);
kl = int64(1);
ku = int64(2);
ab = [complex(0),  0 + 0i,  0 + 0i,  0 + 0i;
      0 + 0i,  0 + 0i,  0.97 - 2.84i,  0.59 - 0.48i;
      0 + 0i,  -2.05 - 0.85i,  -3.99 + 4.01i,  3.33 - 1.04i;
      -1.65 + 2.26i,  -1.48 - 1.75i,  -1.06 + 1.94i,  -0.46 - 1.72i;
      0 + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0 + 0i];
b = [ -1.06 + 21.5i,  12.85 + 2.84i;
      -22.72 - 53.9i,  -70.22 + 21.57i;
      28.24 - 38.6i,  -20.73 - 1.23i;
      -34.56 + 16.73i,  26.01 + 31.97i];
[ab, ipiv, info] = f07br(m, kl, ku, ab);
[bOut, info] = f07bs(trans, kl, ku, ab, ipiv, b)
 

bOut =

  -3.0000 + 2.0000i   1.0000 + 6.0000i
   1.0000 - 7.0000i  -7.0000 - 4.0000i
  -5.0000 + 4.0000i   3.0000 + 5.0000i
   6.0000 - 8.0000i  -8.0000 + 2.0000i


info =

                    0



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

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