How can I write a mathematical function in Matlab?

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ND 2015-6-25

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评论： ND 2015-7-1

采纳的回答： Walter Roberson

Can I build an equation which have different components?

variables written in cell array(1x3) such as 'X1' 'X2' 'X3'

exponents written in matrix (3x3 double) e.g [0 .5 -.5; .5 -.5 0; 1 0 .5]

Parameters written in matrix(3x1) e.g [.55;.25;.6893]

can I get equation like

F= .55*X2^.5*X3^-.5+ .25*X1^.5*X2^-.5+.6893*X1^1*X3^.5

Many thanks

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Answer 1

Walter Roberson 2015-6-25

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在 MATLAB Online 中打开

MATLAB is a general purpose programming language. You can build up any string you want.

subf = sprintf('*X%d^e%d', [1:3; 1:3]);
formula = ['a1' subf '+a2' subf '+a3' subf];

What you should try to avoid is executing such strings. Dynamic generation of code is error prone.

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Walter Roberson 2015-6-26

在 MATLAB Online 中打开

Building up the equation symbolically is different than building it up in a string. When doing it symbolically you can create the template and then subs() the values and variable names in to make it more specific. Or better yet, don't substitute the values in, build it up symbolically as pure variables and calculate the jacobians or differentiate it as pure variables, which is something you could do once on the template and then afterwards substitute in specific values if you ever need to evaluate it.

syms A E V
form = feval(symengine, 'k->sum(A[k]*(product(V[n]^E[k, n], n = 1 .. p))/V[k]^E[k, k], k = 1 .. p)');
syms X1 X2 X3
X = [X1, X2, X3];
form3 = subs(form(length(X)), V, X);   %one specific case for 3
syms X4 X5 X21
X2 = [X2, X4, X5, X21];
form3b = subs(form(length(X2)), V, X2);  %another one in different variables

and you can then do things like diff(form3, X1) and otherwise manipulate it without substituting in the actual values. When it becomes time to put in actual values,

expr = subs(form3, {A, E}, {a, e});
exprb = subs(form3b, {A, E}, {a2, e2});

Walter Roberson 2015-6-27

在 MATLAB Online 中打开

syms A E V
form = evalin(symengine, 'p->sum(A[k]*(product(V[n]^E[k, n], n = 1 .. p))/V[k]^E[k, k], k = 1 .. p)');
X = {'X1', 'X2', 'X3'};
Xs = cellfun(@sym, X);
formX = subs(form(length(X)), V, Xs);

and you can differentiate this and take jacobians to your heart's content. (Hearts' content if you have multiple hearts.) For example,

diff(formX, Xs(2))

When you want to specialize it with particular numbers,

specialized = subs(formX, {A, E}, {a, e});

and you can compute with that result. You can also

fstring = char(specialized);

and that will give you a string containing what you have been asking for (but which is not what you actually need except for display purposes.)

If you wanted to create a MATLAB anonymous objective function for fitting purposes then you could do

ffun = matlabFunction(specialized, 'Vars', Xs);

I do not much concerned that the symbolic mechanism is "more difficult than you expected".

form = evalin(symengine, 'p->sum(A[k]*(product(V[n]^E[k, n], n = 1 .. p))/V[k]^E[k, k], k = 1 .. p)');

is pretty simple and returns a symbolic function that generates the formula for any number of variables. You wanted a sum of products of indexed variables and this expresses a sum of products of indexed variables very compactly. You wanted differentiation and this allows you true differentiation, not trying to fake it by taking numeric differentiation or by doing textual processing on strings to implement the chain rule mechanically.

Walter Roberson 2015-6-28

在 MATLAB Online 中打开

"syms" is for creating symbolic formula. The evalin() creates a symbolic function. When the symbolic function is executed being passed the number of variables involved, called p in the function, the execution of the function creates a template for the desired formula. In the template, the location for the parameter is named A, the location of the variable name is named V, and the location of the exponent is named E. You can substitute particular variable names or particular parameter arrays or exponent arrays into the template by using subs() in the way I show. And when you have the template for a particular number of variables you can substitute different names or values into the template without having to recreate the template.

The expression the symbolic function would create when passed the integer 3 would be

A[1]*V[2]^E[1,2]*V[3]^E[1,3] + A[2]*V[1]^E[2,1]*V[3]^E[2,3] + A[3]*V[1]^E[3,1]*V[2]^E[3,2]

This is the pattern that has to be followed for 3 variables. If you subs() in [X1, X2, X3] for V then the specialized version would become

A[1]*X2^E[1,2]*X3^E[1,3] + A[2]*X1^E[2,1]*X3^E[2,3] + A[3]*X1^E[3,1]*X2^E[3,2]

and if you subs() in [X9, X3, X17] for V then the specialized version for that would become

A[1]*X3^E[1,2]*X17^E[1,3] + A[2]*X9^E[2,1]*X17^E[2,3] + A[3]*X9^E[3,1]*X3^E[3,2]

It is not necessary to recompute what the form of the expression is; it is only necessary to substitute in different names.

Keep in mind that what is being created here is symbolic formula. When you specialize the formula by subs() in the array values you give in your example, and the formula

.55*X2^(.5)*X3^(-.5)+.25*X1^(.5)*X2^(-.5)+.6893*X1^(1)*X3^(.5)

gets created, this is in symbolic form and it is not going to try to immediate execute it based upon whatever value is in the workspace for X1, X2, X3. Requesting that it be executed for particular X1, X2, X3 would be an additional step. The matlabFunction() call that I showed would translate the above symbolic formula into an anonymous MATLAB function

@(X1, X2, X3) .55.*X2.^.5.*X3.^-.5+.25.*X1.^.5.*X2.^-.5+.6893.*X1.^1.*X3.^.5

all ready to be called with specific numeric values at the MATLAB level.

With regards to the /V[k]^E[k, k] part of the formula:

the first term of your sum involves X2, X3 but not X1, the second term involves X1, X3 but not X2, the third term involves X1, X2 but not X3. There are two ways of handling that: one can directly generate "product of all of the variables but leave out the K'th variable"; or one can generate "product of all of the variable and divide that by the K'th variable to get the same result as if the K'th had been left out". It happens that generating all of the product of all of the variables but leaving a particular one out involves more complicated code than multiplying all of the entries together and then dividing by the one that is not wanted. An analogy would be "Calculate the sum of the first N integers but leave out the number M": you can calculate that as

sum(1:M-1)+sum(M+1:N)

or you can calculate it as

sum(setdiff(1:N),M)

or you can calculate it as

sum(1:N)-M

You can see that removing the effect of the M makes for much simpler code than if you had to avoid adding in the value for M.

Notice the whole bit is being evaluated in the symbolic engine. In MATLAB if you want to take the product of values, you have to calculate the values first and then use prod() on them to multiply the terms together. In the symbolic engine you can instead say product(expression,range_of_values) like product((a[k]-1)^2,k=1..3) meaning product([(a[1]-1)^2, (a[2]-1)^2, (a[3]-1)^2]) meaning (a[1]-1)^2*(a[2]-1)^2*(a[3]-1)^2 . This is more compact and intuitive than the way you would have to do it in MATLAB itself, prod(arrayfun(@(V) (V-1)^2, a))

Likewise the symbolic engine supports sum(expression,range_of_values) indicating summation over the values indicated.

The symbolic expression I created is a summation (sum()) of terms of an expression, where the expression is a[k] times the product of terms of an expression involving one variable and one exponent. This is like a nested for loop.

Walter Roberson 2015-6-28

在 MATLAB Online 中打开

A point that I think needs to be made:

You cannot use MATLAB to generate MATLAB code directly.

If you have the symbolic toolbox, you can use MATLAB to generate symbolic formula which can be mathematically manipulated with mathematical reasoning, with the tool knowing for example that diff(x^n,x) is n*x^(n-1), or knowing that cos(theta)^2+sin(theta)^2 = 1 . That tool allows substitutions of values for expressions and subexpressions and the substitutions can be numeric or symbolic expressions. Considerable mathematical reasoning power is available through the symbolic toolbox. The symbolic toolbox includes a routine named matlabFunction() which can convert symbolic formulae into MATLAB anonymous functions.

If you do not have the symbolic toolbox or choose not to use it, then you can generate strings in MATLAB code format. If the string is a single expression, you can use str2fun() to have MATLAB parse the string and convert it to the anonymous function represented by the content of the string. If you choose not to do that or the string represents a command, then you can write the string to a file and then ask MATLAB to execute the file. Or you can use eval() on the string.

Converting a string to an anonymous function using str2fun(), and even more so using eval(), have some hidden risks that can make it easy to destroy the state of your program or even delete files on your system. The risks are notably lower when you use symbolic formula instead of str2fun() or eval(). Using eval() is quite likely to cause difficult to track down trouble; str2fun() and symbolic formula are a lot less likely to lead to the same scenarios.

You need to differentiate between something that is data that might happen to have an interesting form, compared to something is a command. At the MATLAB command prompt, typing in

quit

is going to leave MATLAB. But you can have a string

F = 'tiuq'

which is data and when you use

fliplr(F)

then the result is the string 'quit' but it is still data. Data can be manipulated.

If you have

F2 = str2fun(['@() ', fliplr(F)])

then F2 is still data, but it is data that has been parsed and is ready to be executed. If you now invoke

F2()

then you putting what was formerly data into execution and your MATLAB session is going to execute.

Now imagine what would happen in the code you are thinking of if someone were to say that the variable names to use for this call of the routine should be 'X1, 'X2', and 'quit()' . You "somehow" create

.55*X2^.5*quit()^-.5+.25*X1^.5*X2^-.5+.6893*X1^1*quit()^.5

and then what happens? As long as what you are creating is data then nothing special happens, and if what you are creating is a symbolic formula then you can continue to work with it including differentiation and jacobians because the symbolic engine will treat "quit" as just another variable name. But if you eval() the data at the MATLAB level then you are in trouble.

This is why it is crucial that you understand the difference between code and data, and so understand why I created the formula as data that can then be manipulated in useful ways.

Walter Roberson 2015-6-29

I had no reason at all to expect that the number of variables was fixed. If the number of variables was fixed then there would be no point in generating the formula for the purposes of taking the jacobian or differentiating, because those could be done ahead of time and the numbers substituted in afterwards.

It appears that I was mistaken about only two of the three variables being used; I had not noticed that you had exponents of 0: you had bothered to write in an exponent of 1 but not an exponent of 0. The adjustment would make my symbolic function even simpler.

Anyhow, you still need to decide what form you want the result in. Do you want the result as a string? As a symbolic expression? As an anonymous function? Do you want to be passing in values for everything including values for the variables and have just a pure numeric output? (If so then the names of the variables are irrelevant.) If your answer is "string" then be advised that you cannot differentiate a string or form its jacobian. If your answer is "anonymous function" then be advised that differentiating an anonymous function or forming its jacobian would have to be done as numeric approximation, which would be an inefficient thing to do when you could so easily do an exact differentiation if you worked with a symbolic expression.

Walter Roberson 2015-6-30

在 MATLAB Online 中打开

function form = generateFormula(a, e)
  form = '';
  for k = 1 : length(a)
    thispart = sprintf('%g*X1^%g*X2^%g*X3^%g+', a(k), e(k,1), e(k,2), e(k,3));
    form = [form thispart];
  end
  form(end) = [];  %remove extra '+'
end

and the test code

%generate expression
%Use the "a" and "e" that are in your workspace!
formula = generateFormula(a, e)   %it returns a string now!

I am done. I am not going to answer any more questions on this topic. Do not email me about it.

You now have the code to create the string that you wanted, and what you do with it is your problem.

ND 2015-7-1

Many thanks Walter for all help and I am very sorry for that you are right I should use the code that result symbolic expression . Please delete the last your comment

Many thanks

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Answer 2

Thorsten 2015-6-29

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编辑：Thorsten 2015-6-29

在 MATLAB Online 中打开

If X take some values and you make sure that the e and p parameters are in the order of pairs, e.g., for N = 3

   2
   3
   3

You can use the following code:

e = [0.5 -.5; 0.5 -0.5; 1 0.5];
p = [.55 0.25 0.6893];
X = [345 3 27]; % array of number
% code below should work for arbitrary number N of X values, 
% as long as e is N x 2 and p is 1 x N:
N = numel(p);
pairs = nchoosek(1:N, 2);
F = 0;
for i=1:N
  F = p(i)*X(pairs(i,1))^e(i,1)*X(pairs(i,2))^e(i,2);
end

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Thorsten 2015-6-29

Ok, then it is straightforward

F = p(1)*X(2)^e(1,1)*X(3)^e(1,2) + ...
    p(2)*X(1)^e(2,1)*X(3)^e(2,2) + ...
    p(3)*X(2)^e(3,1)*X(3)^e(3,2);

Walter Roberson 2015-6-29

The variables are strings, Thorsten, and the number of terms (the p values) is variable.

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How can I write a mathematical function in Matlab?

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