Chapter 10, Computer Assisted Proofs and Self-Validating Methods

Siegfried M. Rump

INTLAB

The new Version 7 of INTLAB is available. INTLAB is Matlab toolbox for self-validating algorithms. A number of changes, improvements and speed-ups have been implemented.

It comprises of

interval arithmetic for real and complex data including vectors and matrices (very fast),
interval arithmetic for real and complex sparse matrices (very fast),
automatic differentiation (forward mode, vectorized computations, sparse storage),
- Gradients to solve systems of nonlinear equations,
- Hessians for global optimization,
automatic slopes (sequential approach, slow for many variables),
univariate and multivariate (interval) polynomials,
rigorous real interval standard functions (fast, very accurate, 3 ulps),
rigorous complex interval standard functions (fast, rigorous, but not necessarily sharp inclusions),
rigorous input/output,
accurate summation, dot product and matrix-vector residuals (interpreted, but fairly fast),
multiple precision interval arithmetic with error bounds (does the job, slow), and more.

The philosophy of INTLAB is that *everything* is written in Matlab code to assure best portability.

The new version handles Hessians in pure floating point and with verified bounds. As an example consider a model problem where the function to be minimized is

   function y = f(x)
     N = length(x);
     I = 1:N-4;
     y = sum( (-4*x(I)+3.0).^2 ) + sum( ( x(I).^2 + 2*x(I+1).^2 + ...
       3*x(I+2).^2 + 4*x(I+3).^2 + 5*x(N).^2 ).^2 );

with initial approximation xs=ones(N,1) for N=1000. This means 1e6 elements in the gradient and 1e9 elements in the Hessian, or 8 GByte in full storage. In our implementation Hessians are stored sparse using a special storage scheme allowing efficient computations.

The following is executable code to calculate an inclusion of a stationary point of f by first performing a simple Newton iteration followed by a verification step for the resulting nonlinear system. Error estimations are completely rigorous.

 >> n = 1000;
    xs = ones(n,1);
    tic
    X = verifynlss(@f,xs,'hSparseSPD');
    t = toc
    maxrelerr = max(relerr(X))


    t = 23.8040
    maxrelerr = 5.5992e-013

Inclusions of all components of a stationary point are to some 13 decimal digits and takes 24 seconds on my 800 MHz Pentium III Laptop. Symmetric positive definiteness of the Hessian can be verified as well, ensuring a (local) minimum of f in X.

INTLAB works under Windows, Unix and Mac OS and is tested under Matlab Versions 5.3 to 7.0 . INTLAB is freely available for non-commercial use from

    http://www.ti3.tu-harburg.de/intlab/

For demonstration, also Matlab-implementations of our new summation and dot product routines are given, S. M. Rump, T. Ogita, and S. Oishi Accurate Floating-Point Summation, SIAM J. Sci. Comput. Vol. 31, Issue 2, pp. 1269-1302 (2008).

Comments always welcome. Best wishes

Siegfried M. Rump, Email rump@tu-harburg.de

Date: 23 August 2004 and 4 January 2006 and 17 December 2012

Prof. Dr. Siegfried M. Rump
Inst. f. Computer Science III
Technical University Hamburg-Harburg
Schwarzenbergstr. 95
21071 Hamburg
Germany