Three routines from earlier chapters are tested.

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · Research software practice, verification and validation

Scientific Computing for Physicists, Ch Is the equation solved right? Testing, debugging and code verification

Problem

Three routines from earlier chapters are tested. ((a)) A routine sums the

N=500

diagonal entries of a matrix, each entry equal to

0.1

, so the exact trace is

50

. A unit test compares the routine's result with

50

using the backward-error tolerance

N\epsilon_{\rm mach}/2

for a sequential floating-point sum, with

\epsilon_{\rm mach}=2^{-52}

. Compute the relative tolerance, and say whether the test passes given a measured relative error of

8.8\times10^{-15}

. ((b)) An eigen-decomposition routine on a symmetric

n=100

matrix is tested for the invariant that the sum of the computed eigenvalues equals the trace, checked again after the matrix is transformed by a random orthogonal similarity transform (a rotation of the basis, which leaves the trace unchanged). The property-test tolerance is taken as

10n\epsilon_{\rm mach}/2

. Compute it, and say whether a measured relative deviation of

4\times10^{-15}

passes. ((c)) A routine meant to return the central-difference estimate of

\mathrm{d}(\sin x)/\mathrm{d}x

x=1

, of formal order

p_{\rm formal}=2

, gives errors

e(0.1)=4.29\times10^{-2}

e(0.05)=2.13\times10^{-2}

and

e(0.025)=1.06\times10^{-2}

. Compute

\hat{p}

from each consecutive pair using

\hat{p}=\ln[e(h)/e(h/2)]/\ln2

, say whether the routine is verified, and name a coding slip consistent with the observed order.

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