symbolic evaluation where function is obtained from user input text

Hello,

I'm an undergrad student taking a numerical methods course. The material is fairly dry so i try make it more interesting by implementing each method as a script in matlab - i am a matlab novice. (I'm also lazy so i want to save the effort doing this by hand!). I'm currently trying to code the trapezoidal integration method but i've run into a small problem. So far my code prompts the user to input the function, the number of subintervals to use, the upper and lower bounds of the integral and the number of significant digits the question asks to round to.

prompt = 'input function (of x):  ';
f = input(prompt,'s');
prompt = 'input n:  ';
n = input(prompt);
prompt ='input upper bound b: ';
b = input(prompt);
prompt = 'input lower bound a: ';
a = input(prompt);
prompt = 'enter significant digits required: ';
sig = input(prompt);

display('h is: ')
h = round(((b-a)/(n)),sig);
display(h)
syms x;
actual = double(int(f,x,a,b));
actual = round(actual,sig);

Next i want to calculate the approximation using the Trapezoidal method. I first thought of doing something like this:

T = symsum(0.5(f(xk-1)+f(xk))*h, k, 1, n);

but the user has entered the function as text so f isn't actually a function. I'm looking for some advice on what would be the best way to implement this in matlab? Is there a way for me to turn the user inputted function (eg) x*exp(x)) into a symbolic function? Perhaps a better way would be to create a vector of values for the subintervals and then use a for loop? Any advice would be much appreciated.

[Walter Roberson]

Note: you forgot the multiply between the 0.5 and what follows.

[Stephen23]

"I'm an undergrad student taking a numerical methods course..."

It seems strange to use symbolic operations during a numeric methods course.

Answer 1

[Karan Gill]

Starting R2017b, use str2sym.

>> prompt = 'input function (of x):  ';
f = input(prompt,'s');
input function (of x):  sin(x)

>> syms myfun(x)
>> myfun(x) = str2sym(f)
myfun(x) =
sin(x)

>> fplot(myfun) % for example

Answer 2

[Walter Roberson]

In current versions of MATLAB with the Symbolic Toolbox, it is possible to use

temp = sym(f);
v = symvar(temp);
Fsym = symfun( temp, v)

Now Fsym is a symbolic function

This will give you a warning:

Warning: Support of character vectors will be removed in a future release. Character vectors can be used only for variable names and numbers. Instead, to create
symbolic expressions first create symbolic variables using 'syms'. To evaluate character vectors and strings representing symbolic expressions, use 'str2sym'. 
> In sym>convertExpression (line 1558)
  In sym>convertChar (line 1463)
  In sym>tomupad (line 1213)
  In sym (line 210)

You can disable that with

oldstate = warning('off', 'symbolic:sym:sym:DeprecateExpressions');
temp = sym(f);
warning(oldstate);
v = symvar(temp);
Fsym = symfun( temp, v);

The more general approach is

varlist = strjoin(symvar(f),',');
Fhandle = str2func( ['@(', varlist, ') ', f ]);

Now Fhandle is the function handle of a numeric (non-symbolic) function. You can, if need be,

Fsym = sym(Fhandle);

to produce a symbolic function.

With regards to

T = symsum(0.5(f(xk-1)+f(xk))*h, k, 1, n);

Even if you fix the multiplication and switch to the symbolic function,

T = symsum(0.5*(Fsym(xk-1)+Fsym(xk))*h, k, 1, n);

that will not work. The variable xk does not exist, and if it did exist then because all of the terms would be the same, this would be equivalent to

n * (0.5*(Fsym(xk-1)+Fsym(xk))*h)

You are probably thinking of something like,

    syms k
    T = symsum(0.5*(Fsym(x(k-1))+Fsym(x(k)))*h, k, 2, n);  %(Notice the change to start k from 2)

However, in that expression, k would be a symbolic variable, and it is never permitted to index by a symbolic variable. You need to instead construct the individual terms and then sum() them

Fxk = Fsym(x);
terms = 0.5 * (Fxk(1:end-1) + Fxk(2:end)) * h;
T = sym(terms);

... but if you had stayed with the numeric function handle, you could instead have used

T = trapz(x, arrayfun(Fhandle, x) );