Symbolic expression with matrix input and output

I want to symbolically define a matrix weighted norm like this:

Obviously I do not want to hard code the dimensions of vector and matrix.

I tried to define a symbolic expression like this:

C_mat_g_term_vec = sym('C_g_T_%d%d', [1, 18]);
C_mat_g_term = diag(C_mat_g_term_vec)

g_term_vec = sym('g_term_%d%d', [18, 1]);

% norm_mat_expr = @(v_in, W_in)((eye(size(v_in, 1), size(v_in,1))*v_in)).'*eye(size(W_in, 1))*W_in*(eye(size(v_in), size(v_in,1))*v_in)
norm_mat_expr = @(v_in, W_in)(v_in).'*W_in*(v_in)

norm_mat = sym(norm_mat_expr)

norm_mat(g_term_vec, C_mat_g_term)

Giving the error message:

Error using sym/subsindex (line 825)

Invalid indexing or function definition. Indexing must follow MATLAB indexing. Function arguments must be symbolic variables, and function body must be sym expression.

Error in sym/subsref (line 870)

R_tilde = builtin('subsref',L_tilde,Idx);

So apparently Matlab Sim-Engine handles symbols as scalars.

Does someone has a practical solution for this? :)

The concrete answer is given in the comments below:

C_mat_g_term_vec = sym('C_g_T_%d_%d', [1, 18]);
syms C_mat_g_term
C_mat_g_term = diag(C_mat_g_term_vec);
g_term_vec = sym('g_term_%d_%d', [18, 1]);
norm_mat_expr = @(v_in, W_in)(v_in).'*W_in*(v_in)
norm_mat_expr(g_term_vec, C_mat_g_term)

Answer 1

[Walter Roberson]

The symbolic engine does treat all unresolved symbols as scalars. That is hardcoded into the way it works internally and would be difficult to change. Therefore at the MATLAB level any symbolic function you define must be a function with particular size of arguments.

It looks to me as if you should not sym() the function handle and should instead pass particular arrays to it.

[Hendrik Lorenz]

Do you mean numeric arrays when you say particular arrays?

That is not what I want to do. I want to get a symbolic expression out of my function. If that is not possible in Matlab then I am happily switchting to Python and SymPy, just I am doing the numerics in Matlab so would be nice to also do the analytic part here.

When you say I should not apply sym() to the function handle you say to write the code like this:

clear all
C_mat_g_term_vec = sym('C_g_T_%d,%d', [1, 18]);
syms C_mat_g_term
C_mat_g_term = diag(C_mat_g_term_vec);

g_term_vec = sym('g_term_%d,%d', [18, 1]);

norm_mat_expr = @(v_in, W_in)(v_in).'*W_in*(v_in)

norm_mat_expr(g_term_vec, C_mat_g_term)

This gives me the error:

Error using sym>createCharMatrix (line 1474)

If the second argument specifies the dimensions of the generated symbolic vector or matrix, then the first argument must be a character vector specifying the base name for vector or matrix elements. It must be a valid variable name.

Error in sym>tomupadWithSize (line 1247)

S = createCharMatrix(x, n);

Error in sym (line 237)

S.s = tomupadWithSize(x, n);

[Walter Roberson]

C_mat_g_term_vec = sym('C_g_T_%d_%d', [1, 18]);
syms C_mat_g_term
C_mat_g_term = diag(C_mat_g_term_vec);
g_term_vec = sym('g_term_%d_%d', [18, 1]);
norm_mat_expr = @(v_in, W_in)(v_in).'*W_in*(v_in)
norm_mat_expr(g_term_vec, C_mat_g_term)

[Hendrik Lorenz]

So the problem is the comma here ok.

This works fine, thanks!

[Walter Roberson]

What I meant about passing particular arrays is that if you were to try to define

syms x y
f(x,y) = x*y;

intending matrix multiplication, then you would not be able to do so. In the second statement, MATLAB would resolve x and y each to symbolic variables that are scalars, and it "knows" that scalar times scalar is scalar, so it would treat the * as element-by-element multiplication. If you to later try f(C_mat_g_term_vec, g_term_vec) it would not use matrix multiplication as you might hope: it would do element-by-element multiplication. That would generate an error because implicit expansion is not done for symbolic operations.

In turn this means you cannot develop algorithms such as

syms x y a b
g(x,y) = inv(x) * f(x,y)
h(a, b) = sin(a) + g(a,b)  %here you are passing nominal arrays a and b to g
c = sym('c', [4, 4]);
d = sym('d', [4, 4]);
h(c,d)                     %here you are passing actual arrays c and d to h

and what you have to do instead is

f = @(x,y) x*y
g = @(x,y) inv(x) * f(x,y)
h = @(a,b) sin(a) + g(a,b);
c = sym('c', [4, 4]);
d = sym('d', [4, 4]);
h(c,d)                     %here you are passing actual arrays c and d to h

The actual full-sized symbolic arrays get passed to the function handles, and the function handles preserve that full size into lower layer function handles and preserve algebraic operatons, so the x*y within f will be doing a matrix multiply of two actual arrays instead of two nominal arrays represented by symbols.

[Walter Roberson]

You could define a symbolic function

syms g C
g2c(g,C) = transpose(g) * C * g

but because symbolic variables are treated as scalars, this would come out as C*g.^2

You could define

C = sym('c', [4, 4]);
g = sym('d', [4, 4]);
g2c(reshape([g,C],1,[])) = transpose(g) * C * g

but that would define

g2c(d1_1, d2_1, d3_1, d4_1, d1_2, d2_2, d3_2, d4_2, d1_3, d2_3, d3_3, d4_3, d1_4, d2_4, d3_4, d4_4, c1_1, c2_1, c3_1, c4_1, c1_2, c2_2, c3_2, c4_2, c1_3, c2_3, c3_3, c4_3, c1_4, c2_4, c3_4, c4_4) =

which takes specific sizes of matrix inputs and would not work for (say) 5 x 5

There is no way at the MATLAB level to define a symbolic function that takes "a matrix" as an argument. Function handles bridge the gap.

There are ways inside the symbolic engine to create complete symbolic functions that accept arrays and can deal with them, and if you know the right rituals you can create such a function inside the symbolic engine starting at the MATLAB level -- but MATLAB has no way to display the result, and will crash the symbolic engine when you try to display the result.

[Hendrik Lorenz]

Thanks again for the detailed explanation, much appreciated! Function handles as described work fine for my purposes.