Symbolical differentiation of parametric function

Question

Astrid Barzaghi 2023-7-24

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1999928-symbolical-differentiation-of-parametric-function

评论： Dyuman Joshi 2023-7-25

采纳的回答： Dyuman Joshi

Hello! I am trying to compute the symbolic partial derivative of a parametric function I have defined.

I will report here both main and function

C = [-0.016 1.82e-2 5.5e-3 -2.06e-5 5.91e-7];

G = 6.67259e-20;

R_main = 780e-3;

l = 2;

mass_sys = 5.28e11; %[kg] mass of the system

mass_ratio = 0.0093; % mass ratio

mu = 1 / ((1/mass_ratio) + 1);

m_main = (1 - mu) * mass_sys;

syms x y z

U_main_x = diff(@(x, y, z) U_sph_harm(x, y, z, C, G, m_main, R_main, l), x);

function U = U_sph_harm(x, y, z, C, G, M, R, l)

[lambda, phi, r] = cart2sph(x, y, z);

sum_U = 0;

C_count = 1;

for l_count = 1:l

Plm = legendre(2*l, sin(phi));

for m_count = 0:l_count

sum_U = sum_U + (R/r)^(2*l) * C(C_count) * cos(2 * m * lambda) * Plm(2 * m_count);

C_count = C_count + 1;

end

U = G * M / r * (1 + sum_U);

I get the following error

Conversion to logical from sym is not possible.

Error in legendre (line 101)

if ~isreal(x) || ischar(x)|| max(abs(x(:))) > 1

Error in U_sph_harm (line 8)

Plm = legendre(2*l, sin(phi));

Error in main_asteroid (line 92)

U_main_x = diff(@(x, y, z) U_sph_harm(x, y, z, C, G, m_main, R_main, l), x);

Error in sym>funchandle2ref (line 1601)

S = x(S{:});

Error in sym>tomupad (line 1514)

x = funchandle2ref(x);

Error in sym (line 332)

S.s = tomupad(x);

Error in sym/privResolveArgs (line 1102)

argout{k} = sym(arg);

Error in sym/diff (line 29)

args = privResolveArgs(S, invars{:});

Error in main_asteroid (line 92)

U_main_x = diff(@(x, y, z) U_sph_harm(x, y, z, C, G, m_main, R_main, l), x);

Could you please help me sort out why?

Thank you!

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Answer 1

Dyuman Joshi 2023-7-24

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https://ww2.mathworks.cn/matlabcentral/answers/1999928-symbolical-differentiation-of-parametric-function#answer_1277828

在 MATLAB Online 中打开

The function you are using for Legendre polynomials only accepts numerical inputs.

What you are looking for is legendreP, the symbolic math toolbox function.

However, there are other errors in your code, particularly in the addition line of double for loop -

Plm(2 * m_count)

It's not clear to me what you want do with this line of code.

As you can see below, that Plm is a function of x, y, and z. So, in case you want to find the value of the function, you will need to provide three inputs to it.

C = [-0.016 1.82e-2 5.5e-3 -2.06e-5 5.91e-7];

G = 6.67259e-20;

R_main = 780e-3;

l = 2;

mass_sys = 5.28e11; %[kg] mass of the system

mass_ratio = 0.0093; % mass ratio

mu = 1 / ((1/mass_ratio) + 1);

m_main = (1 - mu) * mass_sys;

syms x y z

U_main_x = diff(U_sph_harm(x, y, z, C, G, m_main, R_main, l), x)

Plm =

Array indices must be positive integers or logical values.

Error in indexing (line 936)
R_tilde = builtin('subsref',L_tilde,Idx);

Error in solution>U_sph_harm (line 22)
sum_U = sum_U + (R/r)^(2*l) * C(C_count) * cos(2 * M * lambda) * Plm(2 * m_count);

function U = U_sph_harm(x, y, z, C, G, M, R, l)

[lambda, phi, r] = cart2sph(x, y, z);

sum_U = 0;

C_count = 1;

for l_count = 1:l

%Corrected function

Plm = legendreP(2*l, sin(phi))

for m_count = 0:l_count

% v correction

sum_U = sum_U + (R/r)^(2*l) * C(C_count) * cos(2 * M * lambda) * Plm(2 * m_count);

C_count = C_count + 1

end

U = G * M / r * (1 + sum_U);

end

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Astrid Barzaghi 2023-7-25

P is the associated legendre function of degree 2l, 2m evaluated in sin(phi).

It turned out that the easiest way to solve the problem was to write myself a function containing the asscoiated legendre functions needed.

The final script is the following

C = [-0.016 1.82e-2 5.5e-3 -2.06e-5 5.91e-7];

G = 6.67259e-20;

R_main = 780e-3;

l = 2;

mass_sys = 5.28e11; %[kg] mass of the system

mass_ratio = 0.0093; % mass ratio

mu = 1 / ((1/mass_ratio) + 1);

m_main = (1 - mu) * mass_sys;

syms x y z

U_main_x = diff(U_sph_harm(x, y, z, C, G, m_main, R_main, l), x);

function U = U_sph_harm(x, y, z, C, G, M, R, l)

[lambda, phi, r] = cart2sph(x, y, z);

sum_U = 0;

C_count = 1;

for l_count = 1:l

for m_count = 0:l_count

l_count_st = num2str(2 * l_count);

m_count_st = num2str(2 * m_count);

lm_st = [l_count_st, m_count_st];

lm = str2double(lm_st);

Plm = ass_legendre(lm);

Plm = Plm(sin(phi));

sum_U = sum_U + (R/r)^(2*l) * C(C_count) * cos(2 * m_count * lambda) * Plm;

C_count = C_count + 1;

end

U = G * M / r * (1 + sum_U);

function Plm = ass_legendre(lm)

switch lm

case 20

Plm = @(x) (1/2) * (3 * x^2 - 1);

case 22

Plm = @(x) 3 * (1 - x^2);

case 40

Plm = @(x) (1/8) * (35 * x^4 - 30 * x^2 + 3);

case 42

Plm = @(x) (15/2) * (7 * x^2 - 1) * (1 - x^2);

case 44

Plm = @(x) 105 * (1 - x^2);

end

Thanks again for the help!

Dyuman Joshi 2023-7-25

在 MATLAB Online 中打开

Ah, I see, it is the Associate Legendre Polynomial. I was not aware of these, that's why I was confused with the notation.

Well you can generate them in this way -

syms x

AP42 = associate(4,2,x)

AP42(x) =

AP20 = associate(2,0,x)

AP20(x) =

function AP = associate(l,m,x)

%Definition taken from Wikipedia

AP(x) = (-1)^m*(1-x^2)^(m/2)*diff(legendreP(l,x),x,m);

end

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Symbolical differentiation of parametric function

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