Institute for Reliable Computing
Head: Prof. Dr. Siegfried M. Rump

Following are some timings on an Intel(R) Xeon(R) CPU E5-1620 with 3.6 GHz using Matlab R2012b on Windows 7 (64 bit) operating system. The source code for the test is here. We start with matrix multiplication and compare the pure floating-point with the verified product, the latter for point data (data without tolerance) and interval data. Note this is comparing apples and oranges: pure floating-point only gives an approximation whereas the latter algorithms provide results which are verified to be correct.

Matrix multiplication, time [sec]

 pure verified verified verified dimension fl-pt A*A A*intA intA*intA 1000 0.028 0.059 0.098 0.13 2000 0.24 0.48 0.71 0.92 5000 3.1 6.1 9.1 12.1 10000 22 45 67 86

INTLAB is optimized for fast execution without sacrifycing the correctness of the results. For example, the routine mtimes.m for matrix multiplication consists of more than 700 lines of Matlab code (without comments). The timing factor is between 2 and 4 compared to the Matlab (pure fl-pt) multiplication.

Next we show timings for dense linear systems. Here an extra-precise residual iteration is available. To stay with our philosophy to use only Matlab code, the latter is a Matlab implementation, suffering from interpretation overhead. Fortunately only matrix-vector multiplications use extra-precise residuals, so the timing factor is between 5 and 10 compared to Matlab's backslash.

Dense linear systems (up to factor 2 faster than previous version), time [sec]

 pure verified verified verified dimension fl-pt A\b high acc. A\b intA\intb 1000 0.022 0.20 0.23 0.28 2000 0.13 1.11 1.14 1.5 5000 1.4 9.7 10.5 16 10000 8.9 62 63 114

Fast and competitive algorithms for sparse linear systems basically exist only for the symmetric definite case. INTLAB applies some symmetric preordering using symamd.m, Matlab's backslash does not. Therefore A\b is very slow without preordering. With preordering the timing factor to pure floating-point is roughly 8. Recall that again we compare apples and oranges.

Sparse s.p.d. linear systems (approx. 10 nonzero elements per row), time [sec]

 pure symand verified verified dimension fl-pt fl-fp A\b intA\intb 1000 0.033 0.0044 0.060 0.034 2000 0.24 0.011 0.13 0.14 5000 3.3 0.058 1.07 1.07 10000 23 0.29 5.0 5.0 20000 - 1.6 22 24 50000 - 19 185 192

Sometimes INTLAB is faster than pure floating-point. There are such examples for quadrature, see demoquad in INTLAB. The following is an optimization problem listed in Coconut. INTLAB solves the problem by finding a root of the nxn nonlinear system ∇ f(x)=0 using the Hessian toolbox. An inclusion of a stationary point is computed, and by verifying the Hessian at this point to be positive definite, it is proved to be a local minimum. Note that in fact the Hessian evaluated at the inclusion of the stationary point is proved to be positive definite. This Hessian is an interval matrix including in particular the Hessian at the stationary point.

 fminsearch local nonlin.system local maximum verification dimension fl-pt [sec] minimum verefied [sec] minimum rel.error pos.def. 50 1.3 3181 0.38 178 7.0e-015 0.042 100 2.8 6733 0.116 379 1.0e-014 0.022 300 19 29342 0.16 1180 1.9e-014 0.028 1000 1243 100655 0.69 3984 3.5e-014 0.096 3000 - - 5.0 11995 6.1e-014 0.64 10000 - - 56 40034 1.1e-013 7.0

As can be seen the Matlab's fminsearch.m is much slower than the verified nonlinear solver. The comparison is not entirely fair because Matlab uses a Nelder-Mead search without derivatives.

Prof. Dr. Siegfried M. Rump
Institute for Reliable Computing
Hamburg University of Technology
Schwarzenbergstr. 95
21071 Hamburg
Germany