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The {{Forge|symbolic|symbolic package}} is part of the | The {{Forge|symbolic|symbolic package}} is part of the [[Octave Forge]] project. | ||
[[Category:Octave Forge]] | |||
[[ | === Demos and usage examples === | ||
[[ | |||
* '''I'm trying to substitute a double value into an expression. How can I avoid getting "warning: Using rat() heuristics for double-precision input (is this what you wanted?)".''' | |||
In general, you should be very careful when converting floating point ("doubles") to symbolic variables, that's why the warning is bothering you. | |||
<source lang="octave"> | |||
## Demo of how to use a number (which was calculated in an octave | |||
## variable) in a symbolic calculation, without getting a warning. | |||
## use octave to calculate some number: | |||
a = pi/2 | |||
## now do some work with the symbolic pkg | |||
syms x | |||
f = x * cos (x) | |||
df = diff (f) | |||
## Now we want to evaluate df at a: | |||
# subs (df, x, a) # this gives the "rats" warning (and gives a symbolic answer) | |||
## So instead, try | |||
dfh = function_handle (df) | |||
dfh (a) | |||
ans = -1.5708 | |||
## And you can evaluate dfh at an array of "double" values: | |||
dfh ([1.23 12.3 pi/2]) | |||
ans = | |||
-0.82502 4.20248 -1.57080 | |||
</source> | |||
* '''Demo of how to graph symbolic functions (by converting SYMBOLIC functions into ANONYMOUS functions)''' | |||
<source lang="octave"> | |||
## The following code will produce the same vector field plot as Figure 1.14 from Example 1.6 (pg. 39) from A Student's Guide to Maxwell's Equations by Dr. Daniel Fleisch. | |||
## Make sure symbolic package is loaded and symbolic variables declared. | |||
pkg load symbolic | |||
syms x y | |||
## Write a Vector Field Equation in terms of symbolic variables | |||
vectorfield = [sin(pi*y/2); -sin(pi*x/2)]; | |||
## Vector components are converted from symbolic into "anonymous functions" which allows them to be graphed. | |||
## The "'vars', [x y]" syntax ensures each component is a function of both 'x' & 'y' | |||
iComponent = function_handle (vectorfield(1), 'vars', [x y]); | |||
jComponent = function_handle (vectorfield(2), 'vars', [x y]); | |||
## Setup a 2D grid | |||
[X,Y] = meshgrid ([-0.5:0.05:0.5]); | |||
figure | |||
quiver (X, Y, iComponent (X, Y), jComponent (X,Y)) | |||
</source> | |||
* '''Demo of Anonymous function to symbolic function and back to anonymous function and then the use of the interval pkg.''' | |||
<source lang="octave"> | |||
% this is just a formula to start with, | |||
% have fun and change it if you want to. | |||
f = @(x) x.^2 + 3*x - 1 + 5*x.*sin(x); | |||
% these next lines take the Anonymous function into a symbolic formula | |||
pkg load symbolic | |||
syms x; | |||
ff = f(x); | |||
% now calculate the derivative of the function | |||
ffd = diff(ff, x); | |||
% and convert it back to an Anonymous function | |||
df = function_handle(ffd) | |||
% this uses the interval pkg to find all the roots between -15 an 10 | |||
pkg load interval | |||
fzero (f, infsup (-15, 10), df) | |||
ans ⊂ 4×1 interval vector | |||
[-5.743488743719015, -5.743488743719013] | |||
[-3.0962279604822407, -3.09622796048224] | |||
[-0.777688831121563, -0.7776888311215626] | |||
[0.22911205809043574, 0.2291120580904359] | |||
</source> | |||
* '''Demo of inputting a function at the input prompt and making an Anonymous function.''' | |||
<source lang="octave"> | |||
# This prog. shows how to take a | |||
# string input and make it into an anonymous function | |||
# this uses the symbolic pkg. | |||
disp("Example input") | |||
disp("x^2 + 3*x - 1 + 5*x*sin(x)") | |||
str_fucn=input("please enter your function ","s") | |||
fucn_sym=sym(str_fucn) | |||
f=function_handle(fucn_sym) | |||
# now back to symbolic | |||
syms x; | |||
ff=formula(f(x)); | |||
% now calculate the derivative of the function | |||
ffd=diff(ff); | |||
% and convert it back to an Anonymous function | |||
df=function_handle(ffd) | |||
% now lets do the second derivative | |||
ffdd=diff(ffd); | |||
ddf=function_handle(ffdd) | |||
% and now plot them all | |||
x1=-2:.0001:2; | |||
plot(x1,f(x1),x1,df(x1),x1,ddf(x1)) | |||
grid minor on | |||
legend("f","f '", "f '' ") | |||
</source> | |||
* '''Demo of ODE with a step input and initial conditions.''' | |||
<source lang="octave"> | |||
## This is a demo of a second order transfer function and a unit step input. | |||
## in laplace we would have 1 1 | |||
## _______________ * _____ | |||
## s^2 + sqrt(2)*s +1 s | |||
## | |||
## So the denominator is s^3 + sqrt(2) * s^2 + s | |||
# and for laplace initial conditions area | |||
## t(0)=0 t'(0) =0 and the step has initial condition of 1 | |||
## so we set t''(0)=1 | |||
## In the code we use diff(y,1)(0) == 0 to do t'(0)=0 | |||
## | |||
## I know that all this can be done using the control pkg | |||
## But I used this to verify that this solution is the | |||
## same as if I used the control pkg. | |||
## With this damping ratio we should have a 4.321% overshoot. | |||
## | |||
syms y(x) | |||
sqrt2=sym(1.41421); | |||
de =diff(y, 3 ) +sqrt2*diff(y,2) + diff(y) == 0; | |||
f = dsolve(de, y(0) == 0, diff(y,1)(0) == 0 , diff(y,2)(0) == 1) | |||
ff=function_handle(rhs(f)) | |||
x1=0:.01:10; | |||
y=ff(x1); | |||
plot(x1,y) | |||
grid minor on | |||
</source> | |||