Figures of merit

This page describes how the various figures of merit can be specified.

• Figures of merit for ordinary lattice rules:

  • the \(\mathcal P_{\alpha}\) discrepancy:
    –figure-of-merit Palpha with \(\alpha\)=alpha,
  • the \(\mathcal R_{\alpha}\) criterion:
    –figure-of-merit Ralpha with \(\alpha\)=alpha,

  For the two last figures of merit, the CU prefix may be added to fasten the computations by using the coordinate-uniform evaluation algorithm when an \(\ell_2\) norm is used.

  • the spectral figure of merit:
    –figure-of-merit spectral;

• Figures of merit for polynomial lattice rules:

  • the \(\mathcal P_{\alpha}\) discrepancy:
    –figure-of-merit Palpha with \(\alpha\)=alpha,
  • the \(\mathcal R\) criterion:
    –figure-of-merit R;

  For both figures of merit, the CU prefix may be added to fasten the computations by using the coordinate-uniform evaluation algorithm when an \(\ell_2\) norm is used.

• Figures of merit for digital nets:
  • the \(\mathcal P_{\alpha}\) discrepancy:
    Only with the coordinate-uniform evaluation algorithm and an \(\ell_2\) norm.
    –figure-of-merit CU:Palpha –norm-type 2 with \(\alpha\)=alpha,
  • the \(\mathcal R\) criterion:
    Only with the coordinate-uniform evaluation algorithm and an \(\ell_2\) norm.
    –figure-of-merit CU:R –norm-type 2,
  • the t-value merit:
    –figure-of-merit t-value,
  • the projection-dependent t-value merit:
    –figure-of-merit projdep:t-value,
  • the projection-dependent t-value based star discrepancy bound merit
    –figure-of-merit projdep:t-value:starDisc,
  • the projection-dependent resolution-gap merit:
    –figure-of-merit projdep:resolution-gap;
• Figures of merit for interlaced polynomial lattice rules and interlaced digital nets:
  • the \(\mathcal I^{(a)}_{\alpha, d}\) interlaced discrepancy:
    Only with an \(\ell_1\) norm.
    –figure-of-merit CU:IAalpha –norm-type 1 with \(\alpha\)=alpha,
  • the \(mathcal I^{(b)}_{\alpha, d}\) interlaced discrepancy:
    Only with an \(\ell_1\) norm. –figure-of-merit CU:IB –norm-type 1.
  • the \(mathcal I^{(c)}_{\alpha, d}\) interlaced discrepancy:
    Only with an \(\ell_1\) norm.
    –figure-of-merit CU:ICalpha –norm-type 1 with \(\alpha\)=alpha.