higham

%REL_ERR_FORMULA  Compare two formulas for relative error.

n = 500;

a = 3;
x = a*ones(n,1,'single');
y = zeros(size(x));
d = eps(single(a));
for i=1:n
    i-n/2

higham

function F = funm_randomized(A,fun)
%FUNM_RANDOMIZED  Evaluate general matrix function using
% randomized approximate diagonalization method of Davies (2007).
tol = 8*eps(A(1,1));  % Tolerance for single or double precision A.
E = randn(size(A)); 
[V,D] = eig(A + (tol*norm(A,'fro')/norm(E,'fro'))*E);
F = V*diag(fun(diag(D)))/V;

higham

%NCM_COMPARE
% M-file to carry out experiment in "Explicit Solutions to Correlation
% Matrix Completion Problems, with an Application to Risk Management and
% Insurance" by Dan I. Georgescu, Nicholas J. Higham and Gareth W. Peters.

C = [%
1     0.25  0.6   0.55  0.65  0     0.4  0.6  0.2    0.3
0.25  1     0     0     0     0    NaN   NaN  NaN    NaN 
0.6   0     1     0.75  0.75  0    NaN   NaN  NaN    NaN 
0.55  0     0.75  1     0.5   0    NaN   NaN  NaN    NaN 

higham

function A = edelman
%EDELMAN  Alan Edelman's matrix for which det is computed as the wrong integer.
%   A = EDELMAN is a 27-by-27 matrix of 1s and -1s for which the
%   MATLAB det function returns an odd integer, though the exact
%   determinant is an even integer.

A = [%
 1  1  1  1 -1 -1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1 -1 -1 -1 -1  1 -1 -1 -1
 1 -1 -1  1 -1  1 -1 -1 -1 -1 -1  1 -1 -1 -1 -1 -1  1 -1 -1 -1 -1  1 -1  1  1  1
-1  1 -1 -1  1  1  1  1  1 -1  1 -1  1  1  1  1 -1 -1  1 -1 -1  1  1 -1  1  1 -1

higham

function fd = complex_step(f,x,h)
%COMPLEX_STEP  Complex step approximation to derivative.
%  fd = COMPLEX_STEP(f,x,h) computes the complex step approximation 
%  fd to the derivative of f at x, using step h (default 1e-100).
if nargin < 3, h = 1e-100; end
fd = imag( f(x + sqrt(-1)*h) )/h;

higham

(use-package hydra
  :config
  (global-set-key (kbd "C-x m") 'hydra-major/body)
  (global-set-key (kbd "<f3>")   'hydra-bib-etc/body)
  (global-set-key (kbd "C-<f3>") 'hydra-dired/body)
)

(defhydra hydra-major (:color blue :columns 4)
  "major-mode"
  ("b" bibtex-mode "bibtex")

higham

% Complex step example.  Requires Symbolic Math Toolbox.

format long
syms x; f = atan(x)/(1+exp(-x^2))

% Derivative at $a = 2$:
a = 2; fd = double(subs( diff(f), a))

% Convert symbolic function to MATLAB function.
f = matlabFunction(f);