# Generate low discrepancy sequences¶

In this examples we are going to expose the available low discrepancy sequences in order to approximate some integrals.

The following low-discrepancy sequences are available:

• Sobol

• Faure

• Halton

• reverse Halton

• Haselgrove

To illustrate these sequences we generate their first 1024 points and compare with the sequence obtained from the pseudo random generator (Merse Twister) as the latter has a higher discrepancy.

```import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt

ot.Log.Show(ot.Log.NONE)
```
1. Sobol sequence

```dimension = 2
size = 1024
sequence = ot.SobolSequence(dimension)
sample = sequence.generate(size)
graph = ot.Graph("Sobol", "", "", True, "")
cloud = ot.Cloud(sample)
view = viewer.View(graph)
``` 1. Halton sequence

```dimension = 2
sequence = ot.HaltonSequence(dimension)
sample = sequence.generate(size)
graph = ot.Graph("Halton", "", "", True, "")
cloud = ot.Cloud(sample)
view = viewer.View(graph)
``` 1. Halton sequence in high dimension: bad filling in upper dimensions

```dimension = 20
sequence = ot.HaltonSequence(dimension)
sample = sequence.generate(size).getMarginal([dimension - 2, dimension - 1])
graph = ot.Graph(
"Halton (" + str(dimension - 2) + "," + str(dimension - 1) + ")",
"dim " + str(dimension - 2),
"dim " + str(dimension - 1),
True,
"",
)
cloud = ot.Cloud(sample)
view = viewer.View(graph)
``` 1. Scrambled Halton sequence in high dimension

```dimension = 20
sequence = ot.HaltonSequence(dimension)
sequence.setScrambling("RANDOM")
sample = sequence.generate(size).getMarginal([dimension - 2, dimension - 1])
graph = ot.Graph(
"Halton (" + str(dimension - 2) + "," + str(dimension - 1) + ")",
"dim " + str(dimension - 2),
"dim " + str(dimension - 1),
True,
"",
)
cloud = ot.Cloud(sample)
view = viewer.View(graph)
``` 1. Reverse Halton sequence

```dimension = 2
sequence = ot.ReverseHaltonSequence(dimension)
sample = sequence.generate(size)
print(
"discrepancy=",
ot.LowDiscrepancySequenceImplementation.ComputeStarDiscrepancy(sample),
)
graph = ot.Graph("Reverse Halton", "", "", True, "")
cloud = ot.Cloud(sample)
view = viewer.View(graph)
``` ```discrepancy= 0.0035074981424325635
```
1. Haselgrove sequence

```dimension = 2
sequence = ot.HaselgroveSequence(dimension)
sample = sequence.generate(size)
graph = ot.Graph("Haselgrove", "", "", True, "")
cloud = ot.Cloud(sample)
view = viewer.View(graph)
``` Compare with uniform random sequence

```distribution = ot.ComposedDistribution([ot.Uniform(0.0, 1.0)] * 2)
sample = distribution.getSample(size)
print(
"discrepancy=",
ot.LowDiscrepancySequenceImplementation.ComputeStarDiscrepancy(sample),
)
graph = ot.Graph("Mersenne Twister", "", "", True, "")
cloud = ot.Cloud(sample) ```discrepancy= 0.03921823278089642