Simplifying with respect to trig identities to avoid dividing by zero

Question

Morten Nissov 2021-2-27

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https://ww2.mathworks.cn/matlabcentral/answers/757799-simplifying-with-respect-to-trig-identities-to-avoid-dividing-by-zero

评论： Morten Nissov 2021-2-27

I need to calculate the jacobian for a set of differential equations and the calculation with respect to one variable is particularly complicated.

A working example

syms x y v a psi omega dt
state = [x; y; v; a; psi; omega];
wt = omega * dt / 2;
hwt = psi + wt;
fx = state + [
    (v*dt + 0.5*a*dt^2)*cos(hwt)*usinc(wt) + a*dt/omega*sin(hwt)*(cos(wt) - usinc(wt));
    (v*dt + 0.5*a*dt^2)*sin(hwt)*usinc(wt) + a*dt/omega*cos(hwt)*(usinc(wt) - cos(wt));
    a * dt;
    0;
    omega * dt;
    0
];
dfx_domega = diff(fx(1), omega); % this is the derivative to be simplified
function [ y ] = usinc( x )
    %un-normalized sinc function
    i=find(x==0);                                                              
    x(i)= 1;      % From LS: don't need this is /0 warning is off                           
    y = sin(x)./(x);                                                     
    y(i) = 1;
end

The problem is that the derivative results in fractions with ω in the denominator. For this case this is the turn rate which can very possibly be zero, as such I would like to manipulate the equations such that it is not in the denominator as far as possible, this is possible in large by applying

.

Is there any way I can apply simplify or rewrite such that the fractions are simplified as such?

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Answer 1

Hernia Baby 2021-2-27

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How about using logical indexing?

function [ y ] = usinc( x )                                                            
    y = sin(x)./(x); % y(x==0) => NaN
    y(x==0) = 1;     % y(x==0) => 1
end

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Morten Nissov 2021-2-27

The usinc function works fine, the problem is the symbolic math produces equations with lots of ω terms in the denominator, which isn't realizable for small values of

, which is very typical.

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