vectorized operations on symbolic functions

Hi, Can we apply vectorized operations on symbolic functions to avoid loops?

syms x1 x2 x3; % symbolic variables
y = x1^3/3 + x2^2/2 - x3; % symbolic function y
X = rand(500,3) % each row representing a combination of x1 x2 x3

I used for loop, but it is taking more time.

y_values = zeros(size(X,1))
for ii = 1:size(X,1)
    y_values(ii) = subs(y, [x1, x2, x3], X(ii,:));
end

I have tried following but since there is inconsistency between sizes of old and new it don't work for me.

y_values = subs(y, [x1, x2, x3], X); % Evaluate the function for all combinations in matrix X

Thanks

Answer 1

[Hassaan]

syms x1 x2 x3; % symbolic variables
y = x1^3/3 + x2^2/2 - x3; % symbolic function y

% Generate a random matrix X with 500 rows and 3 columns
X = rand(500, 3); % each row representing a combination of x1 x2 x3

% Evaluate the symbolic function y for each row in X
% Convert the matrix X to a cell array where each row is a separate cell
X_cell = num2cell(X, 2);

% Use cellfun to apply the subs function to each cell (row) of X_cell
y_values = cellfun(@(row) subs(y, [x1, x2, x3], row), X_cell);

% Convert y_values to a double array if needed
y_values = double(y_values);

disp(y_values(1:10))

   -0.6359
   -0.1162
   -0.2625
   -0.1633
   -0.8502
   -0.9749
    0.4400
   -0.2646
   -0.2648
   -0.2096

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[Muhammad Uzair]

Thank you for your helpful answer! It has improved my understanding of the topic. I am currently working on optimization using a generic algorithm. Unfortunately, my evaluation function is taking too much time for a single iteration, which is becoming unaffordable due to the high time cost. I suspect that most of the time is consumed in finding the Jacobian matrix value from the symbolic function. I have attempted using cell operations, but since the Jacobian is a square matrix (3x6) in my case, it cannot provide a single output value. I have considered decomposing the matrix into 18 functions and finding their value arrays using cells, but this approach seems computationally expensive. Could you please guide me on a more efficient way to handle this?

[Dyuman Joshi]

Note that using num2cell and subsequently cellfun is not efficient.

Answer 2

[Matt J]

syms x1 x2 x3; % symbolic variables
y = x1^3/3 + x2^2/2 - x3; % symbolic function y
X = rand(500,3); % each row representing a combination of x1 x2 x3

ynum=matlabFunction(y) %convert to a numeric function

ynum = function_handle with value:
    @(x1,x2,x3)-x3+x1.^3./3.0+x2.^2./2.0

yvalues=ynum(X(:,1),X(:,2),X(:,3))

yvalues = 500×1

-0.2351 0.0690 -0.4473 -0.5099 -0.6491 -0.5465 -0.7009 -0.3988 0.0451 -0.2412

Answer 3

[Dyuman Joshi]

Better to use a function handle -

x = rand(500,3);
y = @(x) x(:,1).^3/3 + x(:,2).^2/2 - x(:,3);
out1 = y(x)

out1 = 500×1

-0.3891 0.3727 -0.1355 -0.1076 -0.7001 -0.4155 -0.0613 -0.3197 -0.8788 -0.0225

For more information, refer to these documentation pages -

Function Handle

Anonymous Functions

Array vs Matrix Operations

[Muhammad Uzair]

Thank you for your assistance. In my current scenario, I have a square matrix that I need to compute for various input values. I’m unsure whether function handles would be suitable for this case. Additionally, the output matrix will be of size 3x6. Is there a way to obtain individual pages of the output matrix? Specifically, I’m aiming for an output structure of 3x6xn, where ‘n’ represents the number of input combinations.

[Christine Tobler]

Your best approach then might be to make your input variables have the individual pages already as a third dimension:

x = randn(3, 1, 500);
y = randn(1, 6, 500);
z = x.*y;
size(z)

ans = 1×3

3 6 500

The formula here is of course completely made up, it would depend on what your formula for this 3x6 matrix is.

Answer 4

[Aquatris]

Yet another method, you can define your symbolic parameter as a matrix symbolic

n = 500;
X = sym('X', [500 3]);

% I assume here you want a element wise power ('.^') instead of matrix power
y = X(:,1).^3/3 + X(:,2).^2/2 - X(:,3); % symbolic function y

X_val = rand(500,3); % each row representing a combination of x1 x2 x3

y_values = double(subs(y, X, X_val)) % Evaluate the function for all combinations in matrix X

y_values = 500×1

-0.4390 -0.7329 -0.2091 -0.6554 -0.7112 -0.7845 0.1304 -0.5119 -0.1252 -0.3767

size(y_values)

ans = 1×2

500 1

Answer 5

[Walter Roberson]

syms x1 x2 x3; % symbolic variables
y = x1^3/3 + x2^2/2 - x3; % symbolic function y
X = rand(500,3); % each row representing a combination of x1 x2 x3

y_values = double(subs(y, {x1, x2, x3}, {X(:,1), X(:,2), X(:,3)}));

y_values(1:5)

ans = 5×1

-0.5223 0.1043 0.4163 -0.1058 -0.3735

Answer 6

[Torsten]

rng("default")
syms x1 x2 x3; % symbolic variables
y = x1^3/3 + x2^2/2 - x3; % symbolic function y
X = rand(500,3) % each row representing a combination of x1 x2 x3

X = 500x3

0.8147 0.5822 0.6312 0.9058 0.5407 0.3551 0.1270 0.8699 0.9970 0.9134 0.2648 0.2242 0.6324 0.3181 0.6525 0.0975 0.1192 0.6050 0.2785 0.9398 0.3872 0.5469 0.6456 0.1422 0.9575 0.4795 0.0251 0.9649 0.6393 0.4211

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y_values = arrayfun(@(i)double(subs(y,[x1,x2,x3],[X(i,1),X(i,2),X(i,3)])),1:size(X,1))

y_values = 1x500

-0.2814 0.0388 -0.6179 0.0649 -0.5176 -0.5976 0.0616 0.1207 0.3824 0.0827 -0.0344 -0.2115 0.0699 -0.5435 -0.4269 -0.0763 -0.1279 -0.6958 -0.0936 -0.6281 -0.0479 -0.2730 0.1835 -0.0771 0.1207 0.3979 -0.1513 0.1458 -0.5483 -0.3726

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