Difference between revisions of "Symbolic package"

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The {{Forge|symbolic|symbolic package}} is part of the octave-forge project.
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The {{Forge|symbolic|symbolic package}} is part of the [[Octave Forge]] project.
  
[[Category:Octave-Forge]]
+
[[Category:Octave Forge]]
  
 
=== Demos and usage examples ===
 
=== 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?)"
+
* '''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.
 
In general, you should be very careful when converting floating point ("doubles") to symbolic variables, that's why the warning is bothering you.
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  dfh (a)
 
  dfh (a)
 +
 +
ans = -1.5708
 +
  
 
## And you can evaluate dfh at an array of "double" values:
 
## And you can evaluate dfh at an array of "double" values:
  
  dfh ([1.23 42.42 pi/2])
+
  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>
 
</source>
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% have fun and change it if you want to.
 
% have fun and change it if you want to.
  
f=@(x) x.^2 +3*x-1 + 5*x.*sin(x);
+
f = @(x) x.^2 + 3*x - 1 + 5*x.*sin(x);
  
% the next 2 line take the Anonymous function into a symbolic formula
+
% these next lines take the Anonymous function into a symbolic formula
  
 +
pkg load symbolic
 
syms x;
 
syms x;
 
+
ff = f(x);
ff=formula(f(x));
 
  
 
% now calculate the derivative of the function
 
% now calculate the derivative of the function
  
ffd=diff(ff);
+
ffd = diff(ff, x);
  
 
% and convert it back to an Anonymous function
 
% and convert it back to an Anonymous function
  
df=function_handle(ffd)
+
df = function_handle(ffd)
  
  
% this uses the interval pkg. to find all the roots between -15 an 10  
+
% this uses the interval pkg to find all the roots between -15 an 10  
  
 +
pkg load interval
 
fzero (f, infsup (-15, 10), df)
 
fzero (f, infsup (-15, 10), df)
  
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  ##
 
  ##
 
  syms y(x)  
 
  syms y(x)  
  de =diff(y, 3 ) +sqrt(2)*diff(y,2) + diff(y) == 0;
+
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)
 
  f = dsolve(de, y(0) == 0, diff(y,1)(0) == 0 , diff(y,2)(0) == 1)
 
  ff=function_handle(rhs(f))
 
  ff=function_handle(rhs(f))

Latest revision as of 07:41, 10 June 2019

The symbolic package is part of the Octave Forge project.

Demos and usage examples[edit]

  • 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.

## 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


  • Demo of how to graph symbolic functions (by converting SYMBOLIC functions into ANONYMOUS functions)
## 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))


  • Demo of Anonymous function to symbolic function and back to anonymous function and then the use of the interval pkg.
% 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]


  • Demo of inputting a function at the input prompt and making an Anonymous function.
# 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 '' ")


  • Demo of ODE with a step input and initial conditions.
 ## 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