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