Author: Harald Schilly
Views : 611
Compute Environment: Ubuntu 20.04 (Default)

# Octave 6.2.0 on CoCalc Ubuntu 20.04

In :
version()

ans = 6.2.0 
In :
function sqs = squares(n)
# Compute the squares of the numbers from 1 to n.

### BEGIN SOLUTION
# Put correct code here. This code is removed for the student version, but is
# used to confirm that your tests are valid.
if (n <= 0)
error("n must be positive")
endif
sqs = (1:n).^2;
### END SOLUTION
endfunction

In :
# [Modify the tests below for your own problem]
# Check that squares returns the correct output for several inputs:
assert(squares(1), )
assert(squares(2), [1 4])

# Check that squares raises an error for invalid input:
number_of_errors = 0;
for n = [0 -1]
try
squares(n);
catch
number_of_errors++;
end_try_catch
endfor
assert(number_of_errors, 2)

### BEGIN HIDDEN TESTS
# students will NOT see these extra tests
assert(squares(10), [1 4 9 16 25 36 49 64 81 100])
### END HIDDEN TESTS

In :
function s = foo(a, b)
# Compute the sum of a and b.

### BEGIN SOLUTION
s = a + b;
### END SOLUTION
endfunction

In :
foo(23,23)

ans = 46 
In :
[2 3 4]' * [4 3 -1]

ans = 8 6 -2 12 9 -3 16 12 -4 
In :
x = rand(3,3)^3

x = 0.3198 1.3850 1.0445 0.2527 0.9548 0.8514 0.1623 0.8323 0.5163 
In :
save r-octave.mat x -7

In :
scatter(sort(rand(1000, 1)), sort(randn(1000, 1))) In :
i = 0:.1:2*pi;
plot(i, sin(i)) In :
pkg load dicom

error: package dicom is not installed error: called from load_packages at line 47 column 7 pkg at line 588 column 7 
In :
dicomuid()

error: 'dicomuid' undefined near line 1, column 1 The 'dicomuid' function belongs to the dicom package from Octave Forge which seems to not be installed in your system. Please read <https://www.octave.org/missing.html> to learn how you can contribute missing functionality. 
In :
pkg load symbolic;
syms x;
f = sin(x);
diff(f,x)

Symbolic pkg v2.9.0: /usr/local/lib/python3.8/dist-packages/sympy/__init__.py:672: SymPyDeprecationWarning: importing sympy.core.compatibility with 'from sympy import *' has been deprecated since SymPy 1.6. Use import sympy.core.compatibility instead. See https://github.com/sympy/sympy/issues/18245 for more info. self.Warn( Traceback (most recent call last): File "<stdin>", line 1, in <module> File "<stdin>", line 12, in octoutput_drv File "<stdin>", line 54, in octoutput File "<stdin>", line 55, in octoutput File "/usr/local/lib/python3.8/dist-packages/sympy/__init__.py", line 677, in __getattr__ return getattr(self.mod, name) AttributeError: module 'sympy.core.compatibility' has no attribute 'integer_types' Waiting......... Waiting......... 
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pkg load symbolic; syms x
f = 2 * (cos(x) + sin(x)^2)
f1 = diff(f, x)

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xx = -10:0.1:10;
plot(xx, f(xx))

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pkg load image

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a = ones(100, 100);
b = ones(100, 100);
b(3, 1) = .5;
psnr(a, b)


This plot shows the famous 3D sombrero.

A quadratic meshgrid of $x$ and $y$ coordinates is evaluated via $\sqrt{x^2 + y^2} + \epsilon$ and the value $r$ is then the value plotted along the third dimension.

Reference: 3d plots

In [ ]:
tx = ty = linspace (-8, 8, 41)';
[xx, yy] = meshgrid (tx, ty);
r = sqrt (xx .^ 2 + yy .^ 2) + eps;
tz = sin (r) ./ r;
mesh (tx, ty, tz);
xlabel ("tx");
ylabel ("ty");
zlabel ("tz");
title ("3-D Sombrero plot");

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[x,y] = meshgrid(-16:0.5:16);
r = hypot(x,y)/2 + eps;
figure;
surf(sin(r)./r);
colormap(jet);

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This draws the set of points, where the given equation is satisfied. Here, it shows a tilted ellipse.

$x^2 + 3 (y-1)^2 + \frac{x y}{2} = 6$

Reference: ezplot

In [ ]:
ezplot (@(x, y) x.^2 + 3 * (y - 1).^2 + .5 * x .* y - 6)


Imagine you want to evaluate a binary function $f(x,\,y) := x + 2 y$.

For evaluating it in vectorized notation, you need a grid for the cartesian product of all $x$ and $y$.

In :
x = 0:3;
y = 0:4;
[xx, yy] = meshgrid(x, y);
xx + 2*yy

ans = 0 1 2 3 2 3 4 5 4 5 6 7 6 7 8 9 8 9 10 11 

## dsolve and sympy in symbolic

In :
pkg load symbolic
syms y(x)
de = diff(y) == x;
f = dsolve(de, y(1) == 1)

/usr/local/lib/python3.8/dist-packages/sympy/__init__.py:672: SymPyDeprecationWarning: importing sympy.core.compatibility with 'from sympy import *' has been deprecated since SymPy 1.6. Use import sympy.core.compatibility instead. See https://github.com/sympy/sympy/issues/18245 for more info. self.Warn( Traceback (most recent call last): File "<stdin>", line 4, in <module> File "<stdin>", line 12, in octoutput_drv File "<stdin>", line 55, in octoutput File "/usr/local/lib/python3.8/dist-packages/sympy/__init__.py", line 677, in __getattr__ return getattr(self.mod, name) AttributeError: module 'sympy.core.compatibility' has no attribute 'integer_types' error: Python exception: AttributeError: module 'sympy.core.compatibility' has no attribute 'integer_types' occurred while copying variables to Python. Try "sympref reset" and repeat your command? (consider filing an issue at https://github.com/cbm755/octsympy/issues) error: called from pycall_sympy__ at line 191 column 5 valid_sym_assumptions at line 38 column 10 assumptions at line 82 column 7 syms at line 97 column 13 error: 'dsolve' undefined near line 1, column 1 'dsolve' is a method of class 'sym'; it must be called with a 'sym' argument (see 'help @sym/dsolve'). Please read <https://www.octave.org/missing.html> to learn how you can contribute missing functionality. 

## Octave's ODE PKG in Action

In :
pkg load odepkg;

In :
dxdt = @(t, x) - 0.24 * x.^2 + t;
tsteps = [0:0.1:5];
[t, x] = ode45(dxdt, tsteps, [-1:0.5:3]);
plot(t, x) In [ ]:


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You can run numerical optimizations via the optim package.

In this example we minimize the classical Rosenbrock function in 20 dimensions using BFGS.

In :
pkg load optim;

function [obj_value, gradient] = objective(theta, location)
x = theta - location + ones(rows(theta),1); # move minimizer to "location"
[obj_value, gradient] = rosenbrock(x);
endfunction

dim = 20;                 # dimension of Rosenbrock function
theta0 = zeros(dim+1,1);  # starting values
location = (0:dim)/dim;   # true values
location = location';
control = {Inf,1};        # maxiters, verbosity

bfgsmin("objective", {theta0, location}, control);

------------------------------------------------ bfgsmin final results: 65 iterations function value: 2.17563e-16 STRONG CONVERGENCE Function conv 1 Param conv 1 Gradient conv 1 used numeric gradient param gradient (n) change 0.00000 0.00000 -0.00000 0.05000 -0.00000 0.00000 0.10000 0.00000 0.00000 0.15000 -0.00000 0.00000 0.20000 -0.00000 0.00000 0.25000 0.00000 0.00000 0.30000 -0.00000 0.00000 0.35000 -0.00000 0.00000 0.40000 -0.00000 0.00000 0.45000 0.00000 -0.00000 0.50000 0.00000 -0.00000 0.55000 -0.00000 0.00000 0.60000 0.00000 -0.00000 0.65000 0.00000 -0.00000 0.70000 -0.00000 0.00000 0.75000 -0.00000 0.00000 0.80000 0.00000 0.00000 0.85000 0.00000 0.00000 0.90000 0.00000 0.00000 0.95000 -0.00000 0.00000 1.00000 0.00000 0.00000 
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