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The Matrix Computation Toolbox

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The Matrix Computation Toolbox

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11 Sep 2002 (Updated )

A collection of M-files for carrying out various numerical linear algebra tasks.

gfpp(T, c)
function A = gfpp(T, c)
%GFPP   Matrix giving maximal growth factor for Gaussian elim. with pivoting.
%       GFPP(T) is a matrix of order N for which Gaussian elimination
%       with partial pivoting yields a growth factor 2^(N-1).
%       T is an arbitrary nonsingular upper triangular matrix of order N-1.
%       GFPP(T, C) sets all the multipliers to C  (0 <= C <= 1)
%       and gives growth factor (1+C)^(N-1) - but note that for T ~= EYE
%       it is advisable to set C < 1, else rounding errors may cause
%       computed growth factors smaller than expected.
%       GFPP(N, C) (a special case) is the same as GFPP(EYE(N-1), C) and
%       generates the well-known example of Wilkinson.

%       Reference:
%       N. J. Higham and D. J. Higham, Large growth factors in
%          Gaussian elimination with pivoting, SIAM J. Matrix Analysis and
%          Appl., 10 (1989), pp. 155-164.
%       N. J. Higham, Accuracy and Stability of Numerical Algorithms,
%         Second edition, Society for Industrial and Applied Mathematics,
%         Philadelphia, PA, 2002; sec. 9.4.

if ~isequal(T,triu(T)) | any(~diag(T))
   error('First argument must be a nonsingular upper triangular matrix.')
end

if nargin == 1, c = 1; end

if c < 0 | c > 1
   error('Second parameter must be a scalar between 0 and 1 inclusive.')
end

m = length(T);
if m == 1    % Handle the special case T = scalar
   n = T;
   m = n-1;
   T = eye(n-1);
else
   n = m+1;
end

A = zeros(n);
L = eye(n) - c*tril(ones(n), -1);
A(:,1:n-1) = L*[T; zeros(1,n-1)];
theta = max(abs(A(:)));
A(:,n) = theta * ones(n,1);
A = A/theta;

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