The Vegas method, selected with $\mathrm{method}=\mathrm{\_CubaVegas}$, uses importance sampling as a variance reduction technique. It is particularly suitable if the integrand is separable; that is, if it can be written as a product of univariate functions in the integration variables. This can be viewed as the most basic of the four Cuba methods.

The Suave method, selected with $\mathrm{method}=\mathrm{\_CubaSuave}$, uses importance sampling combined with a globally adaptive subdivision strategy. This method is unique to Cuba. It is quite effective, but uses more memory than most other methods.

The Divonne method, selected with $\mathrm{method}=\mathrm{\_CubaDivonne}$, uses stratified sampling for variance reduction. In particular, it attempts to find the minimum and maximum in each region using optimization methods, and attempts to partition the integration region so that all subregions, $r$, have similar values of a quantity called the spread, $s\left(r\right)$, defined by:

where $V\left(r\right)$ is the volume of $r$. The algorithm proceeds in (up to) three phases. In phase 1, the domain of integration is partitioned and estimates for the required number of samples are made. In phase 2, these samples are taken and a ${\mathrm{\chi }}^2$ test assesses whether the estimated accuracy conforms with the results of phase 1. Subregions where it does not undergo a further phase 3 for refinement.

The Cuhre method, selected with $\mathrm{method}=\mathrm{\_CubaCuhre}$, is a deterministic integration method. It is most suitable for moderate dimensions. A different implementation of the same algorithm is available with $\mathrm{method}=\mathrm{\_cuhre}$.

The Cuba library allows the user to specify a large variety of options. These can be passed to the routines using the $\mathrm{methodoptions}=\mathrm{molist}$ argument. Some options are common to all methods, others only occur for some.

If you set ${\mathrm{infolevel}}_{\mathrm{evalf}}$ or ${\mathrm{infolevel}}_{\mathrm{evalf/int}}$, you can obtain some diagnostic information.

$\mathrm{integrand1}≔\frac{1}{1+\mathrm{abs}\left(x-0.2\right)+\mathrm{abs}\left(y-0.7\right)+\mathrm{abs}\left(z+0.3\right)}$

$\mathrm{integrand1}≔\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

$\mathrm{region1}≔\left[x=-1..1\,y=-1..1\,z=-1..1\right]$

$\mathrm{region1}≔\left[x=\mathrm{-1}..1\,y=\mathrm{-1}..1\,z=\mathrm{-1}..1\right]$

The default method is the NAG Cuhre algorithm.

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand1}\,\mathrm{region1}\,'\mathrm{\epsilon }'=0.00001\right)\right)$

Control_multi: integrating on [-1, -1, -1] .. [1, 1, 1] the integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

trying DCUHRE (TOMS algorithm 698)

cuhre: epsrel=.1e-4; minpts=0; maxpts=90000000

cuhre: result=3.05062989428920384

cuhre: abserr=.305046087432878584e-4; usedpts=7806563

result=3.05062989428920384

$3.050629894$

We apply the four Cuba algorithms, with standard options.

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand1}\,\mathrm{region1}\,'\mathrm{\epsilon }'=0.00001\,'\mathrm{method}=\mathrm{\_CubaVegas}'\right)\right)$

Control_multi: integrating on [-1, -1, -1] .. [1, 1, 1] the integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: transformed original integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: with lower bounds [-1., -1., -1.] and upper bounds [1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

$\frac{8.}{1+\left|2.x-1.2\right|+\left|2.y-1.7\right|+\left|2.z-0.7\right|}$

cuba: integration completed successfully

cuba: # of integrand evaluations: 16065000

cuba: estimated (absolute) error: 3.04443e-05

cuba: chi-square probability that the error is not reliable: 0

$3.05063294319739$

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand1}\,\mathrm{region1}\,'\mathrm{\epsilon }'=0.00001\,'\mathrm{methodoptions}=\left[\mathrm{nnew}=50000\right]'\,'\mathrm{method}=\mathrm{\_CubaSuave}'\right)\right)$

Control_multi: integrating on [-1, -1, -1] .. [1, 1, 1] the integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: transformed original integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: with lower bounds [-1., -1., -1.] and upper bounds [1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

$\frac{8.}{1+\left|2.x-1.2\right|+\left|2.y-1.7\right|+\left|2.z-0.7\right|}$

cuba: integration completed successfully

cuba: # of integrand evaluations: 2050000

cuba: estimated (absolute) error: 3.03368e-05

cuba: chi-square probability that the error is not reliable: 1

cuba: number of regions that the domain was divided into: 41

$3.05063339209162$

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand1}\,\mathrm{region1}\,'\mathrm{\epsilon }'=0.00001\,'\mathrm{method}=\mathrm{\_CubaDivonne}'\right)\right)$

Control_multi: integrating on [-1, -1, -1] .. [1, 1, 1] the integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: transformed original integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: with lower bounds [-1., -1., -1.] and upper bounds [1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

$\frac{8.}{1+\left|2.x-1.2\right|+\left|2.y-1.7\right|+\left|2.z-0.7\right|}$

cuba: integration completed successfully

cuba: # of integrand evaluations: 103867

cuba: estimated (absolute) error: 3.04855e-05

cuba: chi-square probability that the error is not reliable: 0

cuba: number of regions that the domain was divided into: 30

$3.05063267628703$

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand1}\,\mathrm{region1}\,'\mathrm{\epsilon }'=0.00001\,'\mathrm{method}=\mathrm{\_CubaCuhre}'\right)\right)$

Control_multi: integrating on [-1, -1, -1] .. [1, 1, 1] the integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: transformed original integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: with lower bounds [-1., -1., -1.] and upper bounds [1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

$\frac{8.}{1+\left|2.x-1.2\right|+\left|2.y-1.7\right|+\left|2.z-0.7\right|}$

cuba: integration completed successfully

cuba: # of integrand evaluations: 200787

cuba: estimated (absolute) error: 3.04852e-05

cuba: chi-square probability that the error is not reliable: 1.11022e-16

cuba: number of regions that the domain was divided into: 791

$3.05039469433604$

We use a separable integrand.

$\mathrm{integrand2}≔\exp \left(\mathrm{add}\left(-\frac{i}{10}\mathrm{abs}\left(x\left[i\right]-\frac{i}{20}\right)\,i=5..15\right)\right)$

$\mathrm{integrand2}≔{\ⅇ}^{-\frac{\left|x_5-\frac{1}{4}\right|}{2}-\frac{3\left|x_6-\frac{3}{10}\right|}{5}-\frac{7\left|x_7-\frac{7}{20}\right|}{10}-\frac{4\left|x_8-\frac{2}{5}\right|}{5}-\frac{9\left|x_9-\frac{9}{20}\right|}{10}-\left|x_{10}-\frac{1}{2}\right|-\frac{11\left|x_{11}-\frac{11}{20}\right|}{10}-\frac{6\left|x_{12}-\frac{3}{5}\right|}{5}-\frac{13\left|x_{13}-\frac{13}{20}\right|}{10}-\frac{7\left|x_{14}-\frac{7}{10}\right|}{5}-\frac{3\left|x_{15}-\frac{3}{4}\right|}{2}}$

$\mathrm{region2}≔\left[\mathrm{seq}\left(x\left[i\right]=0..1\,i=5..15\right)\right]$

$\mathrm{region2}≔\left[x_5=0..1\,x_6=0..1\,x_7=0..1\,x_8=0..1\,x_9=0..1\,x_{10}=0..1\,x_{11}=0..1\,x_{12}=0..1\,x_{13}=0..1\,x_{14}=0..1\,x_{15}=0..1\right]$

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand2}\,\mathrm{region2}\,'\mathrm{\epsilon }'=0.00001\,'\mathrm{method}=\mathrm{\_CubaVegas}'\right)\right)$

Control_multi: integrating on [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] .. [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] the integrand

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: transformed original integrand

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: with lower bounds [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.] and upper bounds [1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: integration completed successfully

cuba: # of integrand evaluations: 3751000

cuba: estimated (absolute) error: 5.86145e-07

cuba: chi-square probability that the error is not reliable: 2.22045e-16

$0.0588897585546207$

We see that about 4000000 function evaluations were needed. If we restrict the number of function evaluations to substantially less, we will not get a floating point result.

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand2}\,\mathrm{region2}\,'\mathrm{\epsilon }'=0.00001\,'\mathrm{methodoptions}=\left[\mathrm{maximalpoints}={10}^6\right]'\,'\mathrm{method}=\mathrm{\_CubaVegas}'\right)\right)$

Control_multi: integrating on [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] .. [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] the integrand

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: transformed original integrand

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: with lower bounds [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.] and upper bounds [1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: could not attain requested accuracy

cuba: # of integrand evaluations: 1007500

cuba: estimated (absolute) error: 1.31319e-06

cuba: chi-square probability that the error is not reliable: 0

evalf/int: error from Control_multi was:
"could not attain requested accuracy; try increasing epsilon or absepsilon or maximalpoints"

${\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}\left({\int }_{0.}^{1.}{\ⅇ}^{-0.5000000000\left|x_5-0.2500000000\right|-0.6000000000\left|x_6-0.3000000000\right|-0.7000000000\left|x_7-0.3500000000\right|-0.8000000000\left|x_8-0.4000000000\right|-0.9000000000\left|x_9-0.4500000000\right|-1.\left|x_{10}-0.5000000000\right|-1.100000000\left|x_{11}-0.5500000000\right|-1.200000000\left|x_{12}-0.6000000000\right|-1.300000000\left|x_{13}-0.6500000000\right|-1.400000000\left|x_{14}-0.7000000000\right|-1.500000000\left|x_{15}-0.7500000000\right|}\ⅆx_5\right)\ⅆx_6\right)\ⅆx_7\right)\ⅆx_8\right)\ⅆx_9\right)\ⅆx_{10}\right)\ⅆx_{11}\right)\ⅆx_{12}\right)\ⅆx_{13}\right)\ⅆx_{14}\right)\ⅆx_{15}$

The integrand is not smooth, so we could try to turn off smoothing. However, in this example it does not make a difference.

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand2}\,\mathrm{region2}\,'\mathrm{\epsilon }'=0.00001\,'\mathrm{methodoptions}=\left[\mathrm{smooth}=\mathrm{false}\right]'\,'\mathrm{method}=\mathrm{\_CubaVegas}'\right)\right)$

Control_multi: integrating on [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] .. [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] the integrand

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: transformed original integrand

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: with lower bounds [0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.] and upper bounds [1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

${\ⅇ}^{-\frac{\left|\mathrm{x[5]}-\frac{1}{4}\right|}{2}-\frac{3\left|\mathrm{x[6]}-\frac{3}{10}\right|}{5}-\frac{7\left|\mathrm{x[7]}-\frac{7}{20}\right|}{10}-\frac{4\left|\mathrm{x[8]}-\frac{2}{5}\right|}{5}-\frac{9\left|\mathrm{x[9]}-\frac{9}{20}\right|}{10}-\left|\mathrm{x[10]}-\frac{1}{2}\right|-\frac{11\left|\mathrm{x[11]}-\frac{11}{20}\right|}{10}-\frac{6\left|\mathrm{x[12]}-\frac{3}{5}\right|}{5}-\frac{13\left|\mathrm{x[13]}-\frac{13}{20}\right|}{10}-\frac{7\left|\mathrm{x[14]}-\frac{7}{10}\right|}{5}-\frac{3\left|\mathrm{x[15]}-\frac{3}{4}\right|}{2}}$

cuba: integration completed successfully

cuba: # of integrand evaluations: 3751000

cuba: estimated (absolute) error: 5.86145e-07

cuba: chi-square probability that the error is not reliable: 2.22045e-16

$0.0588897585546207$

The parameter one will most typically need to adapt for the Suave method is $\mathrm{nnew}$. Lower values mean the process of integration is stopped relatively frequently to assess if further subdivisions are necessary. Also, there will be more subdivisions for a given number of integrand evaluations if $\mathrm{nnew}$ is low. Hence, for low requested accuracy and highly variable integrands, lower values of $\mathrm{nnew}$ are better. For higher requested accuracy and for a relatively uniform integrand, low values induce more subdivisions than necessary and higher values are more suitable.

In the first example above, using the default value for $\mathrm{nnew}$ would be quite slow with the default number of $\mathrm{maximalpoints}$. However, if we lower the requested accuracy, it does work.

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand1}\,\mathrm{region1}\,'\mathrm{\epsilon }'=0.0001\,'\mathrm{method}=\mathrm{\_CubaSuave}'\right)\right)$

Control_multi: integrating on [-1, -1, -1] .. [1, 1, 1] the integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: transformed original integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: with lower bounds [-1., -1., -1.] and upper bounds [1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

$\frac{8.}{1+\left|2.x-1.2\right|+\left|2.y-1.7\right|+\left|2.z-0.7\right|}$

cuba: integration completed successfully

cuba: # of integrand evaluations: 371000

cuba: estimated (absolute) error: 0.000304326

cuba: chi-square probability that the error is not reliable: 1

cuba: number of regions that the domain was divided into: 371

$3.04985592113939$

Still, with $\mathrm{nnew}$ set to 3000 (rather than its default of 1000), we get the result with substantially fewer integrand evaluations.

$\mathrm{evalf}\left(\mathrm{Int}\left(\mathrm{integrand1}\,\mathrm{region1}\,'\mathrm{\epsilon }'=0.0001\,'\mathrm{method}=\mathrm{\_CubaSuave}'\,'\mathrm{methodoptions}=\left[\mathrm{nnew}=3000\right]'\right)\right)$

Control_multi: integrating on [-1, -1, -1] .. [1, 1, 1] the integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: transformed original integrand

$\frac{1}{1+\left|x-0.2\right|+\left|y-0.7\right|+\left|z+0.3\right|}$

cuba: with lower bounds [-1., -1., -1.] and upper bounds [1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

$\frac{8.}{1+\left|2.x-1.2\right|+\left|2.y-1.7\right|+\left|2.z-0.7\right|}$

cuba: integration completed successfully

cuba: # of integrand evaluations: 48000

cuba: estimated (absolute) error: 0.000297572

cuba: chi-square probability that the error is not reliable: 1

cuba: number of regions that the domain was divided into: 16

$3.05050182266965$

The following integrand has substantial spikes. Therefore integration works a little better if we set $\mathrm{flatness}$ to a low number. If we would need higher accuracy, we could determine the most effective value for $\mathrm{flatness}$ at low accuracy, then do the final computation with that value of $\mathrm{flatness}$. (The calling sequence int(..., numeric, ...), used here, is the same as the one with evalf(Int(...)).)

$\mathrm{spikes}≔\mathrm{Statistics}:-\mathrm{Sample}\left(\mathrm{Uniform}\left(0\,1\right)\,\left[6\,4\right]\right)$

$\mathrm{spikes}≔\left[\right]$

$\mathrm{integrand3}≔\mathrm{add}\left(\mathrm{mul}\left(\ln \left(1.7\mathrm{abs}\left(x\left[i\right]-\mathrm{spikes}\left[i,j\right]\right)\right)\,i=1..6\right)\,j=1..4\right)$

$\mathrm{integrand3}≔\ln \left(1.7\left|x_1-0.814723686393179\right|\right)\ln \left(1.7\left|x_2-0.905791937075619\right|\right)\ln \left(1.7\left|x_3-0.126986816293506\right|\right)\ln \left(1.7\left|x_4-0.913375856139019\right|\right)\ln \left(1.7\left|x_5-0.632359246225410\right|\right)\ln \left(1.7\left|x_6-0.0975404049994095\right|\right)+\ln \left(1.7\left|x_1-0.278498218867048\right|\right)\ln \left(1.7\left|x_2-0.546881519204984\right|\right)\ln \left(1.7\left|x_3-0.957506835434298\right|\right)\ln \left(1.7\left|x_4-0.964888535199277\right|\right)\ln \left(1.7\left|x_5-0.157613081677548\right|\right)\ln \left(1.7\left|x_6-0.970592781760616\right|\right)+\ln \left(1.7\left|x_1-0.957166948242946\right|\right)\ln \left(1.7\left|x_2-0.485375648722841\right|\right)\ln \left(1.7\left|x_3-0.800280468888800\right|\right)\ln \left(1.7\left|x_4-0.141886338627215\right|\right)\ln \left(1.7\left|x_5-0.421761282626275\right|\right)\ln \left(1.7\left|x_6-0.915735525189067\right|\right)+\ln \left(1.7\left|x_1-0.792207329559554\right|\right)\ln \left(1.7\left|x_2-0.959492426392903\right|\right)\ln \left(1.7\left|x_3-0.655740699156587\right|\right)\ln \left(1.7\left|x_4-0.0357116785741896\right|\right)\ln \left(1.7\left|x_5-0.849129305868777\right|\right)\ln \left(1.7\left|x_6-0.933993247757551\right|\right)$

$\mathrm{region3}≔\left[\mathrm{seq}\left(x\left[i\right]=0..1\,i=1..6\right)\right]$

$\mathrm{region3}≔\left[x_1=0..1\,x_2=0..1\,x_3=0..1\,x_4=0..1\,x_5=0..1\,x_6=0..1\right]$

$\mathrm{int}\left(\mathrm{integrand3}\,\mathrm{region3}\,'\mathrm{numeric}'\,'\mathrm{\epsilon }'=0.001\,'\mathrm{method}=\mathrm{\_CubaSuave}'\,'\mathrm{methodoptions}=\left[\mathrm{flatness}=1\,\mathrm{nnew}=10000\right]'\right)$

Control_multi: integrating on [0, 0, 0, 0, 0, 0] .. [1, 1, 1, 1, 1, 1] the integrand

$\ln \left(1.7\left|\mathrm{x[1]}-0.814723686393179\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.905791937075619\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.126986816293506\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.913375856139019\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.632359246225410\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.0975404049994095\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.278498218867048\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.546881519204984\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.957506835434298\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.964888535199277\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.157613081677548\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.970592781760616\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.957166948242946\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.485375648722841\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.800280468888800\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.141886338627215\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.421761282626275\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.915735525189067\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.792207329559554\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.959492426392903\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.655740699156587\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.0357116785741896\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.849129305868777\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.933993247757551\right|\right)$

cuba: transformed original integrand

$\ln \left(1.7\left|\mathrm{x[1]}-0.814723686393179\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.905791937075619\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.126986816293506\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.913375856139019\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.632359246225410\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.0975404049994095\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.278498218867048\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.546881519204984\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.957506835434298\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.964888535199277\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.157613081677548\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.970592781760616\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.957166948242946\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.485375648722841\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.800280468888800\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.141886338627215\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.421761282626275\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.915735525189067\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.792207329559554\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.959492426392903\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.655740699156587\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.0357116785741896\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.849129305868777\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.933993247757551\right|\right)$

cuba: with lower bounds [0., 0., 0., 0., 0., 0.] and upper bounds [1., 1., 1., 1., 1., 1.], to the following integrand to be integrated over the unit n-cube:

$\ln \left(1.7\left|\mathrm{x[1]}-0.814723686393179\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.905791937075619\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.126986816293506\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.913375856139019\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.632359246225410\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.0975404049994095\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.278498218867048\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.546881519204984\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.957506835434298\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.964888535199277\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.157613081677548\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.970592781760616\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.957166948242946\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.485375648722841\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.800280468888800\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.141886338627215\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.421761282626275\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.915735525189067\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.792207329559554\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.959492426392903\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.655740699156587\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.0357116785741896\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.849129305868777\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.933993247757551\right|\right)$

cuba: integration completed successfully

cuba: # of integrand evaluations: 4440000

cuba: estimated (absolute) error: 0.00152791

cuba: chi-square probability that the error is not reliable: 1

cuba: number of regions that the domain was divided into: 444

$1.52943666651863$

This method is highly customizable. Consider, for example, the previous integrand.

$\mathrm{int}\left(\mathrm{integrand3}\,\mathrm{region3}\,'\mathrm{numeric}'\,'\mathrm{\epsilon }'=0.001\,'\mathrm{method}=\mathrm{\_CubaDivonne}'\right)$

Control_multi: integrating on [0, 0, 0, 0, 0, 0] .. [1, 1, 1, 1, 1, 1] the integrand

$\ln \left(1.7\left|\mathrm{x[1]}-0.814723686393179\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.905791937075619\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.126986816293506\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.913375856139019\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.632359246225410\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.0975404049994095\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.278498218867048\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.546881519204984\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.957506835434298\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.964888535199277\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.157613081677548\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.970592781760616\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.957166948242946\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.485375648722841\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.800280468888800\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.141886338627215\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.421761282626275\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.915735525189067\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.792207329559554\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.959492426392903\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.655740699156587\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.0357116785741896\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.849129305868777\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.933993247757551\right|\right)$

cuba: transformed original integrand

$\ln \left(1.7\left|\mathrm{x[1]}-0.814723686393179\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.905791937075619\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.126986816293506\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.913375856139019\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.632359246225410\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.0975404049994095\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.278498218867048\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.546881519204984\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.957506835434298\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.964888535199277\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.157613081677548\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.970592781760616\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.957166948242946\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.485375648722841\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.800280468888800\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.141886338627215\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.421761282626275\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.915735525189067\right|\right)+\ln \left(1.7\left|\mathrm{x[1]}-0.792207329559554\right|\right)\ln \left(1.7\left|\mathrm{x[2]}-0.959492426392903\right|\right)\ln \left(1.7\left|\mathrm{x[3]}-0.655740699156587\right|\right)\ln \left(1.7\left|\mathrm{x[4]}-0.0357116785741896\right|\right)\ln \left(1.7\left|\mathrm{x[5]}-0.849129305868777\right|\right)\ln \left(1.7\left|\mathrm{x[6]}-0.933993247757551\right|\right)$