I apologize if I'm asking something trivial. I know that MATLAB's symsum (of the Symbolic Math Toolbox) can be used for series. For instance,
syms k
symsum(2^k/factorial(k),k,1,Inf) % output exp(2) - 1
The series whose general term is
is a divergent one. When I try symsum with this series I get
symsum(log(k+1)/k,k,1,Inf)
That is, MATLAB returns the series unevaluated. However, I wanted a behavior similar to Mathematica's.
Sum[Log[r + 1]/r, {r, 1, Infinity}]
(*Output: During evaluation of In[1]:= Sum::div: Sum does not converge.*)
Is it possible to get such an output from MATLAB's Symbolic Math Toolbox?
Related
I'm using octave 3.8.1 which works like matlab.
I have an array of thousands of values I've only included three groupings as an example below:
(amp1=0.2; freq1=3; phase1=1; is an example of one grouping)
t=0;
amp1=0.2; freq1=3; phase1=1; %1st grouping
amp2=1.4; freq2=2; phase2=1.7; %2nd grouping
amp3=0.8; freq3=5; phase3=1.5; %3rd grouping
The Octave / Matlab code below solves for Y so I can plug it back into the equation to check values along with calculating values not located in the array.
clear all
t=0;
Y=0;
a1=[.2,3,1;1.4,2,1.7;.8,5,1.5]
for kk=1:1:length(a1)
Y=Y+a1(kk,1)*cos ((a1(kk,2))*t+a1(kk,3))
kk
end
Y
PS: I'm not trying to solve for Y since it's already solved for I'm trying to solve for Phase
The formulas located below are used to calculate Phase but I'm not sure how to put it into a for loop that will work in an array of n groupings:
How would I write the equation / for loop for finding the phase if I want to find freq=2.5 and amp=.23 and the phase is unknown I've looked online and it may require writing non linear equations which I'm not sure how to convert what I'm trying to do into such an equation.
phase1_test=acos(Y/amp1-amp3*cos(2*freq3*pi*t+phase3)/amp1-amp2*cos(2*freq2*pi*t+phase2)/amp1)-2*freq1*pi*t
phase2_test=acos(Y/amp2-amp3*cos(2*freq3*pi*t+phase3)/amp2-amp1*cos(2*freq1*pi*t+phase1)/amp2)-2*freq2*pi*t
phase3_test=acos(Y/amp3-amp2*cos(2*freq2*pi*t+phase2)/amp3-amp1*cos(2*freq1*pi*t+phase1)/amp3)-2*freq2*pi*t
Image of formula below:
I would like to do a check / calculate phases if given a freq and amp values.
I know I have to do a for loop but how do I convert the phase equation into a for loop so it will work on n groupings in an array and calculate different values not found in the array?
Basically I would be given an array of n groupings and freq=2.5 and amp=.23 and use the formula to calculate phase. Note: freq will not always be in the array hence why I'm trying to calculate the phase using a formula.
Ok, I think I finally understand your question:
you are trying to find a set of phase1, phase2,..., phaseN, such that equations like the ones you describe are satisfied
You know how to find y, and you supply values for freq and amp.
In Matlab, such a problem would be solved using, for example fsolve, but let's look at your problem step by step.
For simplicity, let me re-write your equations for phase1, phase2, and phase3. For example, your first equation, the one for phase1, would read
amp1*cos(phase1 + 2 freq1 pi t) + amp2*cos(2 freq2 pi t + phase2) + amp3*cos(2 freq3 pi t + phase3) - y = 0
Note that ampX (X is a placeholder for 1, 2, 3) are given, pi is a constant, t is given via Y (I think), freqX are given.
Hence, you are, in fact, dealing with a non-linear vector equation of the form
F(phase) = 0
where F is a multi-dimensional (vector) function taking a multi-dimensional (vector) input variable phase (comprised of phase1, phase2,..., phaseN). And you are looking for the set of phaseX, where all of the components of your vector function F are zero. N.B. F is a shorthand for your functions. Therefore, the first component of F, called f1, for example, is
f1 = amp1*cos(phase1+...) + amp2*cos(phase2+...) + amp3*cos(phase3+...) - y = 0.
Hence, f1 is a one-dimensional function of phase1, phase2, and phase3.
The technical term for what you are trying to do is find a zero of a non-linear vector function, or find a solution of a non-linear vector function. In Matlab, there are different approaches.
For a one-dimensional function, you can use fzero, which is explained at http://www.mathworks.com/help/matlab/ref/fzero.html?refresh=true
For a multi-dimensional (vector) function as yours, I would look into using fsolve, which is part of Matlab's optimization toolbox (which means I don't know how to do this in Octave). The function fsolve is explained at http://www.mathworks.com/help/optim/ug/fsolve.html
If you know an approximate solution for your phases, you may also look into iterative, local methods.
In particular, I would recommend you look into the Newton's Method, which allows you to find a solution to your system of equations F. Wikipedia has a good explanation of Newton's Method at https://en.wikipedia.org/wiki/Newton%27s_method . Newton iterations are very simple to implement and you should find a lot of resources online. You will have to compute the derivative of your function F with respect to each of your variables phaseX, which is very simple to compute since you're only dealing with cos() functions. For starters, have a look at the one-dimensional Newton iteration method in Matlab at http://www.math.colostate.edu/~gerhard/classes/331/lab/newton.html .
Finally, if you want to dig deeper, I found a textbook on this topic from the society for industrial and applied math: https://www.siam.org/books/textbooks/fr16_book.pdf .
As you can see, this is a very large field; Newton's method should be able to help you out, though.
Good luck!
I have been working on solving some equation in a more complicated context. However, I want to illustrate my question through the following simple example.
Consider the following two functions:
function y=f1(x)
y=1-x;
end
function y=f2(x)
if x<0
y=0;
else
y=x;
end
end
I want to solve the following equation: f1(x)=f2(x). The code I used is:
syms x;
x=solve(f1(x)-f2(x));
And I got the following error:
??? Error using ==> sym.sym>notimplemented at 2621
Function 'lt' is not implemented for MuPAD symbolic objects.
Error in ==> sym.sym>sym.lt at 812
notimplemented('lt');
Error in ==> f2 at 3
if x<0
I know the error is because x is a symbolic variable and therefore I could not compare x with 0 in the piecewise function f2(x).
Is there a way to fix this and solve the equation?
First, make sure symbolic math is even the appropriate solution method for your problem. In many cases it isn't. Look at fzero and fsolve amongst many others. A symbolic method is only needed if, for example, you want a formula or if you need to ensure precision.
In such an old version of Matlab, you may want to break up your piecewise function into separate continuous functions and solve them separately:
syms x;
s1 = solve(1-x^2,x) % For x >= 0
s2 = solve(1-x,x) % For x < 0
Then you can either manually examine or numerically compare the outputs to determine if any or all of the solutions are valid for the chosen regime – something like this:
s = [s1(double(s1) >= 0);s2(double(s2) < 0)]
You can also take advantage of the heaviside function, which is available in much older versions.
syms x;
f1 = 1-x;
f2 = x*heaviside(x);
s = solve(f1-f2,x)
Yes, the Heaviside function is 0.5 at zero – this gives it the appropriate mathematical properties. You can shift it to compare values other than zero. This is a standard technique.
In Matlab R2012a+, you can take advantage of assumptions in addition to the normal relational operators. To add to #AlexB's comment, you should convert the output of any logical comparison to symbolic before using isAlways:
isAlways(sym(x<0))
In your case, x is obviously not "always" on one side or the other of zero, but you may still find this useful in other cases.
If you want to get deep into Matlab's symbolic math, you can create piecewise functions using MuPAD, which are accessible from Matlab – e.g., see my example here.
I'm trying to write a generic function for finding the cosine of a value inputted into the function. The formula for cosine that I'm using is:
n
cosx = sum((-1)^n*x^(2n)/(2n)!)
n=1
I've looked at the matlab documentation and this page implies that the "sum" function should be able to do it so I tried to test it by entering:
sum(x^n, n=1..3)
but it just gives me "Error: The expression to the left of the equals sign is not a valid target for an assignment".
Is summing an infinite series something that matlab is able to do by default or do I have to simulate it using a function and loops?
Well if you want to approximate it to a finite number of terms you can do it in Matlab without toolboxes or loops:
sumCos = #(x, n)(sum(((-1).^(0:n)).*(x.^(2*(0:n)))./(factorial(2*(0:n)))));
and then use it like this
sumCos(pi, 30)
The first parameter is the angle, the second is the number of terms you want to take the series to (i.e. effects the precision). This is a numerical solution which I think is really what you're after.
btw I took the liberty of correcting your initial sum, surely n must start from 0 if you are trying to approximate cos
If you want to understand my formula (which surely you do) then you need to read up on some essential Matlab basics namely the colon operator and then the concept of using . to perform element-wise operations.
In MATLAB itself, no, you cannot solve an infinite sum. You would have to estimate it as you suggested. The page you were looking at is part of the Symbolic Math toolbox, which is an add-on to MATLAB. In particular, you were looking at MuPAD, which is rather similar to Mathematica. It is a symbolic math workspace, whereas MATLAB is more of a numeric math workspace. If you own the Symbolic Math toolbox, you can either use MuPAD as you tried to above, or you can use the symsum function from within MATLAB itself to carry out sums of series.
I have a function with 2 symbolic variables which is very complicated and long. I want to have it plotted in a surface. The function looks like this:
y^(1/2)*x - y^(1/2)*(x - 1)*((40*y^2 + 60*y^(1/2) - 60*y^(3/2) - 10)/(90*y^(1/2)) + ...
Whenever I try to plot, I get: "Undefined function 'plotfunc3d' for input arguments of type 'sym'" or "Conversion to double from sym is not possible".
How can I plot a surface? Thanks.
As the error indicates, you're trying to plot a symbolic equation using functions designed for Matlab's default floating-point datatypes. Your question is terse and you didn't even indicate how you're trying to plot the function in question, so I'll just give suggestions.
You have two options. You can use a plot function designed for symbolic math or you can substitute in floating point values for all of your parameters and values. Here is a list of ez- plotting methods that can be used for symbolic equations.
Secondly, you can use the subs function to substitute in values, or vectors/matrices of values into a symbolic equation. You can also try using double if you end up with an equation that is stil symbolic but not in terms of any variables, e.g., double(sym('pi')).
I can't really be more specific because your question wasn't, but you can also try Googling "Matlab plot symbolic function" for more results.
I am working on a function that will generate polynomial interpolants for a given set of ordered pairs. You currently input the indexes of the node points in one vector, and the values of the function to be interpolated in a second vector. I then generate a symbolic expression for the Lagrange polynomial that interpolates that set of points. I would like to be able to go from this symbolic form to a vector form for comparison with test functions and such. That is, I have something that generates some polynomial P(x) in terms of some symbolic variable x. I would like to then sample this polynomial to a vector, and get values for the polynomial over (for example) linspace(-1,1,1000). If this is possible, how do I do it?
I guess I'll include the code that I have so far:
function l_poly = lpoly(x,f)
% Returns the polynomial interpolant as computed by lagrange's formula
syms a
n=size(x,2);
l_poly_vec = 1;
l_poly=0;
for k=1:n,
for l=1:n,
if (k ~= l)
l_poly_vec=l_poly_vec*(a-x(l))/(x(k)-x(l));
end
end
l_poly=l_poly+f(k)*l_poly_vec;
l_poly_vec = 1;
end
I plan on adding a third (or possibly fourth) input depending on how I can solve this issue. I'm guessing I would just need the length of the vector I want to sample to and the endpoints.
If I understand you correctly, you've constructed a Lagrange interpolating polynomial using the symbolic toolbox and now wish to evaluate it over a vector of values. One way to do this is to use the function sym2poly to extract the coefficients of the symbolic polynomial, and then use polyval to evaluate it. Alternatively, you could use matlabFunction to convert your symbolic expression into a regular Matlab function; or use subs to substitute in a numeric value for 'x'.
However, you would probably be better off avoiding the symbolic toolbox altogether and directly constructing the coefficients of the Lagrange interpolating polynomial, or, better yet, use a different interpolation scheme altogether. The function interp1 might be a good place to start.