matlab - How to recursively fill an array with functions? -

so i'm trying write function generate hermite polynomials , it's doing super crazy ... why generate different elements h when start different n? inputting hpoly(2,1) gives

h = [ 1, 2*y, 4*y^2 - 2]

while hpoly(3,1) ,

h = [ 1, 2*y, 4*y^2 - 4, 2*y*(4*y^2 - 4) - 8*y]

( (4y^2 - 2) vs (4y^2 - 4) third element here )

also, can't figure out how evaluate expression. tried out = subs(h(np1),y,x) did nothing.

code:

function out = hpoly(n, x) clc;  syms y      np1 = n + 1;     h = [1, 2*y];     f(np1)      function f(np1)         if numel(h) < np1              f(np1 - 1)              h(np1) = 2*y*h(np1-1) - 2*(n-1)*h(np1-2);         end     end h y = x; out = h(np1);  end 

-------------------------- edit ----------------------------

so got around using while loop instead. wonder why other way didn't work ... (and still can't figure out how evaluate expression other plug in x beginning ... suppose that's not important, still nice know...)

sadly, code isn't fast hermiteh :( wonder why.

function out = hpoly(n, x)      h = [1, 2*x];     np1 = n + 1;      while np1 > length(h)         h(end+1) = 2*x*h(end) - 2*(length(h)-1)*h(end-1);     end  out = h(end)  end 

why code slower? recursion not of matlab's fortes may have improved using recurrence relation. however, hermiteh written in c , loop won't fast because you're using while instead of for , needlessly reallocating memory instead of preallocating it. hermiteh may use lookup table first coefficients or might benefit vectorization using explicit expression. might rewrite function this:

function h = hpoly(n,x) % n - increasing sequence of integers starting @ 0 % x - point @ evaluate polynomial, numeric or symbolic value mx = max(n); h = cast(zeros(1,mx),class(x)); % use zeros(1,mx,'like',x) in newer versions of matlab h(1) = 1; if mx > 0     h(2) = 2*x; end = 2:length(n)-1     h(i+1) = 2*x*h(i)-2*(i-1)*h(i-1); end 

you can call with

syms x; deg = 3; h = hpoly(0:deg,x) 

which returns [ 1, 2*x, 4*x^2 - 2, 2*x*(4*x^2 - 2) - 8*x] (use expand on output if want). unfortunately, won't faster if x symbolic.

if you're interested in numeric results of the polynomial evaluated @ particular values, it's best avoid symbolic math altogether. function above valued double precision x 3 4 orders of magnitude faster symbolic x. example:

x = pi; deg = 3; h = hpoly(0:deg,x) 

yields

h =     1.0e+02 *     0.010000000000000   0.062831853071796   0.374784176043574   2.103511015993210 

note:

the hermiteh function r2015a+, assuming still have access symbolic math toolbox , matlab version r2012b+, can try calling mupad's orthpoly::hermite. hermiteh used function under hood. see here details on how call mupad functions matlab. function bit simpler in returns single term. using for loop:

syms x; deg = 2; h = sym(zeros(1,deg+1)); = 1:deg+1     h(i) = feval(symengine,'orthpoly::hermite',i-1,x); end 

alternatively, can use map vectorize above:

deg = 2; h = feval(symengine,'map',0:deg,'n->orthpoly::hermite(n,x)'); 

both return [ 1, 2*x, 4*x^2 - 2].