examples

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Gosl Examples

See benchmark examples here

Summary

  • Compute (fast) discrete Fourier transform
  • Generate and draw a NURBS toroidal surface
  • Generating normally distributed pseudo-random numbers
  • Visualising Triplets for sparse matrix input
  • Solution of sparse linear system
  • Solution of sparse linear system with complex numbers
  • Numerical differentiation
  • Drawing iso-surfaces with VTK
  • Plotting a contour
  • Root finding problems
  • B-splines: curve, control, and basis
  • Orthogonal polynomials
  • Lagrange interpolation using Chebyshev points
  • Fourier interpolation
  • Solving ordinary differential equations

Compute (fast) discrete Fourier transform

Source code: fun_fft01.go

Generate and draw a NURBS toroidal surface

Source code: gm_nurbs03.go

Generating normally distributed pseudo-random numbers

The rnd package is a wrapper to Go rand package but has some more high level functions to assist on works involving random numbers and probability distributions.

By using the package rnd, it's very easy to generate pseudo-random numbers sampled from a normal distribution.

Source code: rnd_normalDistribution.go

Normally distributed pseudo-random numbers

Visualising Triplets for sparse matrix input

Triplets are the easiest way to input sparse matrix data. They can be visualised using Vismatrix after writing a .smat file.

Given the following matrix:

      _            _
     |  0  2  0  0  |
 a = |  1  0  4  0  |
     |  0  0  0  5  |
     |_ 0  3  0  6 _|

The matrix can be generated using: la_triplet01.go

The matrix can be visualised using Vismatrix resulting in:

Output of Vismatrix

Solution of sparse linear system

Solution of real and sparse linear system using Umfpack and high-level routines.

A small linear system is solved with Umfpack. The sparse matrix representation is initialised with a triplet.

Given the following matrix:

      _                      _
     |  2   3    0    0    0  |
     |  3   0    4    0    6  |
 A = |  0  -1   -3    2    0  |
     |  0   0    1    0    0  |
     |_ 0   4    2    0    1 _|

and the following vector:

      _      _
     |    8   |
     |   45   |
 b = |   -3   |
     |    3   |
     |_  19  _|

solve:

         A.x = b

Output:

a =
    2    3    0    0    0
    3    0    4    0    6
    0   -1   -3    2    0
    0    0    1    0    0
    0    4    2    0    1
b = 8 45 -3 3 19 
x = 0.9999999999999998 2 3 4 4.999999999999998

Source code: la_HLsparseReal01.go

Alternatively, the low-level routines can be used.

In this case, three steps must be taken:

  1. Initialise solver
  2. Perform factorisation
  3. Solve problem

See: la_sparseReal01.go

Solution of sparse linear system with complex numbers

Solution of complex and sparse linear system using Umfpack and high-level routines.

Given the following matrix of complex numbers:

      _                                                  _
     |  19.73    12.11-i      5i        0          0      |
     |  -0.51i   32.3+7i    23.07       i          0      |
 A = |    0      -0.51i    70+7.3i     3.95    19+31.83i  |
     |    0        0        1+1.1i    50.17      45.51    |
     |_   0        0          0      -9.351i       55    _|

and the following vector:

      _                  _
     |    77.38+8.82i     |
     |   157.48+19.8i     |
 b = |  1175.62+20.69i    |
     |   912.12-801.75i   |
     |_     550-1060.4i  _|

solve:

         A.x = b

the solution is:

      _            _
     |     3.3-i    |
     |    1+0.17i   |
 x = |      5.5     |
     |       9      |
     |_  10-17.75i _|

Source code: la_HLsparseComplex01.go

Alternatively, the low-level routines can be used.

See: la_sparseComplex01.go

Numerical differentiation

There are numerous uses for numerical differentiation.

In this example, numerical differentiation is employed to check that the implementation of the derivatives of the sin function is corrected.

Source code: num_deriv01.go

Sin function

Output:

                 x              analytical               numerical                   error
dy/dx   @ 0.000000                       1      0.9999999999999998   2.220446049250313e-16
d²y/dx² @ 0.000000                      -0                       0                       0
dy/dx   @ 0.628319      0.8090169943749473      0.8090169943746159  3.3140157285060923e-13
d²y/dx² @ 0.628319     -0.5877852522924731     -0.5877852522897387  2.7344793096517606e-12
dy/dx   @ 1.256637     0.30901699437494745     0.30901699437699115   2.043698543729988e-12
d²y/dx² @ 1.256637     -0.9510565162951535     -0.9510565163025483   7.394751477818318e-12
dy/dx   @ 1.884956    -0.30901699437494734     -0.3090169943750832   1.358357870628879e-13
d²y/dx² @ 1.884956     -0.9510565162951536     -0.9510565162929511   2.202571458553848e-12
dy/dx   @ 2.513274     -0.8090169943749475     -0.8090169943687026   6.244893491214043e-12
d²y/dx² @ 2.513274     -0.5877852522924732     -0.5877852522882455   4.227729277772596e-12
dy/dx   @ 3.141593                      -1     -0.9999999999784639  2.1536106231678787e-11
d²y/dx² @ 3.141593 -1.2246467991473515e-16                       0  1.2246467991473515e-16
dy/dx   @ 3.769911     -0.8090169943749475     -0.8090169943913278  1.6380341527622022e-11
d²y/dx² @ 3.769911       0.587785252292473      0.5877852522905287  1.9443335830260366e-12
dy/dx   @ 4.398230    -0.30901699437494756    -0.30901699437612157  1.1740053373898718e-12
d²y/dx² @ 4.398230      0.9510565162951535      0.9510565162973443  2.1908030944928214e-12
dy/dx   @ 5.026548     0.30901699437494723     0.30901699436647384   8.473388657392888e-12
d²y/dx² @ 5.026548      0.9510565162951536      0.9510565162933304   1.823208251039432e-12
dy/dx   @ 5.654867      0.8090169943749473       0.809016994400035  2.5087709687454662e-11
d²y/dx² @ 5.654867      0.5877852522924732      0.5877852523075159   1.504263380525117e-11
dy/dx   @ 6.283185                       1      0.9999999999840412   1.595878984517185e-11
d²y/dx² @ 6.283185   2.449293598294703e-16                       0   2.449293598294703e-16

Drawing iso-surfaces with VTK

An isosurface is a geometric construction representing a 2D region containing equal values. This surface is drawn in the 3D space (although the concept can be extended to hyperisosurfaces too) for a given scalar field (i.e. a level).

In this example, the functions to generate families of surfaces resembling a cone and an ellipse are developed. These functions are computed over a 3D grid that is used by VTK to locate the regions of equal values. Two auxiliary scalars fields, p and q, are firstly defined.

Source code: vtk_isosurf01.go

Iso-surface

Plotting a contour

The plt subpackage is a convenient wrapper to python.matplotlib/pyplot that can generate nice graphs. For example:

Source code: plt_contour01.go

Contour

Root finding problems

Example: find the root of

    y(x) = x³ - 0.165 x² + 3.993e-4

within [0, 0.11]. We have to make sure that the root is bounded otherwise Brent's method doesn't work.

Using Brent's method: num_brent01.go

Output:

  it                      x                   f(x)                    err
                                                                  1.0e-14
   0  1.100000000000000e-01 -2.662000000000001e-04  5.500000000000000e-02
   1  6.600000000000000e-02 -3.194400000000011e-05  3.300000000000000e-02
   2  6.044444444444443e-02  1.730305075445823e-05  2.777777777777785e-03
   3  6.239640011030302e-02 -1.676981032316081e-07  9.759778329292944e-04
   4  6.237766369176578e-02 -7.323468182796403e-10  9.666096236606754e-04
   5  6.237758151338346e-02  3.262039076357137e-15  4.108919116063703e-08
   6  6.237758151374950e-02  0.000000000000000e+00  4.108900814037142e-08

x      = 0.0623775815137495
f(x)   = 0
nfeval = 8
niter. = 6

Simple root finding problem solved by Brent's method.

Using Newton's method: num_newton01.go

Output:

  it                    Ldx                 fx_max
                  (1.0e-04)              (1.0e-09)
   0  0.000000000000000e+00  2.778000000000000e-04
   1  3.745954692556634e+06  5.421253067129628e-05
   2  6.176571448942142e+05  1.391803634400563e-06
   2  1.515117884960284e+04  5.314115983194589e-10
. . . converged with fx_max. nit=2, nFeval=4, nJeval=3

x      = 0.062377521883073835
f(x)   = 5.314115983194589e-10
nfeval = 4
niter. = 2

Simple root finding problem solved by Newton's method.

B-splines: curve, control, and basis

Source code: gm_bspline02.go

Orthogonal polynomials

This example generates the following orthogonal polynomials and plot according to figures in [1]

  • Jacobi
  • Chebyshev First Kind
  • Chebyshev Second Kind
  • Legendre
  • Hermite

Source code: fun_orthopoly01.go

Jacobi: Figure 22.1

Jacobi orthogonal polynomials. Figure 22.1 of [1]

Chebyshev1: Figure 22.6

Chebyshev 1st kind orthogonal polynomials. Figure 22.6 of [1]

Chebyshev2: Figure 22.7

Chebyshev 2nd kind orthogonal polynomials. Figure 22.7 of [1]

Legendre: Figure 22.8

Legendre orthogonal polynomials. Figure 22.8 of [1]

Hermite: Figure 22.10

Hermite orthogonal polynomials. Figure 22.10 of [1]

Reference:

[1] Abramowitz M, Stegun IA (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce, NIST

Lagrange interpolation using Chebyshev points

This example interpolates the Runge curve using the following formula:

                  N
      X          ————             X
     I {f}(x) =  \     f(x[i]) ⋅ ℓ (x)
      N          /                i
                 ————
                 i = 0

where ℓ is the Lagrange (cardinal) function.

Source code: fun_laginterp01.go

Generating (nodal) polynomial

With N = 8

               N
       X      ━━━━
      ω (x) = ┃  ┃ (x - X[i])
      N+1     ┃  ┃
             i = 0

Runge curve for many polynomial degrees

Error estimate

Fourier interpolation

Interpolte the boxcar function using truncate Fourier series

Source code: fun_fourierinterp01.go

Solving ordinary differential equations

Hairer-Wanner VII-p2 Eq.(1.1)

References:
  [1] Hairer E, Nørsett SP, Wanner G (1993). Solving Ordinary Differential Equations I:
      Nonstiff Problems. Springer Series in Computational Mathematics, Vol. 8, Berlin,
      Germany, 523 p.
  [2] Hairer E, Wanner G (1996). Solving Ordinary Differential Equations II: Stiff and
      Differential-Algebraic Problems. Springer Series in Computational Mathematics,
      Vol. 14, Berlin, Germany, 614 p.

Source code: ode_hweq11.go

Documentation

Overview

Package examples holds a number of examples. NOTE: this is NOT a 'package' actually. This file is created just to satisfy some development tools such as go vet and eslint

Source Files

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