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NOTE: For the sake of convenience, we are (slowly) moving the examples to another repository: https://github.com/cpmech/gosl-examples
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The examples about Machine Learning have been moved already.
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After installing Gosl, most of the examples below can be executed using
./all.bashSome few examples need extra data files that are available here.
Nonetheless, these examples are not executed by all.bash automatically.
Thus, you may not need to worry about the extra files!
If you want to execute the extra examples, an environment variable named GOSLDATA pointing to the
location of the extra data files is required. For instance:
export GOSLDATA=$HOME/GoslDataFilesSource code: opt_comparison01.go
Source code: opt_conjgrad01.go
Source code: opt_ipm01.go
Source code: opt_ipm02.go
This example reads the MNIST subset data given in Prof. Andrew Ng's course.
The data file is available here
and the environment variable GOSLDATA must be defined as explained above.
Source code: h5_ang_mnist01.go
Source code: fun_fft01.go
Source code: gm_nurbs03.go
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
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:
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
In this case, three steps must be taken:
- Initialise solver
- Perform factorisation
- Solve problem
See: la_sparseReal01.go
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
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
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
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
The plt subpackage is a convenient wrapper to python.matplotlib/pyplot that can generate nice
graphs. For example:
Source code: plt_contour01.go
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
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
Source code: gm_bspline02.go
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
[1] Abramowitz M, Stegun IA (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce, NIST
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
With N = 8
N
X ━━━━
ω (x) = ┃ ┃ (x - X[i])
N 1 ┃ ┃
i = 0
Interpolte the boxcar function using truncate Fourier series
Source code: fun_fourierinterp01.go
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
Source code: plt_bigo.go
Source code: tri_delaunay01.go
Source code: tri_generate01.go
