Linked Questions

131
votes
8answers
8k views

Can one identify the design patterns of Mathematica?

... or are they unnecessary in such a high-level language? I've been thinking about programming style, coding standards and the like quite a bit lately, the result of my current work on a mixed .Net/...
9
votes
4answers
3k views

Implementing Picard's Iteration for solving ODEs

Picard's Iteration is a way of solving the IVP $$y'(x)=f(x,y(x)), \quad y(x_0)=y_0 $$ It consists of defining the following sequence of functions recursively: $$y_0(x):=y_0 \\ y_{n}(x):=y_0+\int_{...
7
votes
4answers
444 views

Recursive piecewise integral relation with piecewise base case?

How is this recursive formula $$ f_{n+1}(z) = \int_0^1 f_{n}(z-y)\,{\rm d}y $$ implemented in Mathematica? The base case is $$ f_1(z) = \begin{cases} 1 & 0\leq z\leq 1 \\ 0 & \text{...
4
votes
4answers
541 views

Memoization of a function defined by a recurrence relation [duplicate]

I have a function which is defined by the following recurrence relation $$h_{n}(x)=h_{n-1}(x)+\frac{\mathrm{e}^{-x^2}}{2^n n!}H_{n}(x)H_{n-1}(x)$$ with the initial condition $h_{0}(x)=0$ and where the ...
168
votes
3answers
15k views

Performance tuning in Mathematica?

What performance tuning tricks do you use to make a Mathematica application faster? MATLAB has an amazing profiler, but from what I can tell, Mathematica has no similar functionality.
18
votes
3answers
3k views

Why does Mathematica use [[ ]] notation for array indexing?

I am confused by why Mathematica uses [[3]] to get the 3rd element, or [[i,j] to get the i,j-th element of a 2D array. This ...
5
votes
3answers
1k views

recursive integration

I am trying to do multiple integrations recursively. For instance, I would like to do the following equation for arbitrary integer $n$: $\displaystyle R_n(t) = \int_0^t \mathrm dt' R_0(t-t') R_{n-1}(...
4
votes
3answers
474 views

How to generate a recurrent sequence

How to generate this type of sequence? $$ f(n, x) = x f'(n-1, x) \hspace{2 mm}, f(0, x) = e^x$$ How do I evaluate it for numerical values for $x = 1$ or any number?
1
vote
3answers
492 views

List of Tribonacci Polynomials with Mathematica? [duplicate]

I want to list top ten of Tribonacci polynomials. I have following algorithm, but it doesnt work. ...
11
votes
2answers
1k views

Dynamic Programming with delayed evaluation

By using dynamical programming, we can save intermediate steps for recursive relations, as in f[n_]:= f[n] = f[n-1] + f[n-2] However, this only stores ...
9
votes
2answers
286 views

using memoization with conditional statement

I would like some advice about the use of memoization in conjunction with conditional statements. Let me try to explain my problem: I am constructing a function depending on a number of parameters. ...
7
votes
2answers
1k views

Why does Expand not work within a function?

I'm writing this fairly simple function: ...
7
votes
2answers
3k views

Series of piecewise functions

Let $f_{0}(x):[0,1]\to[0,1]$ be defined by $$f_{0}(x):=\begin{cases} 3x, & \text{if } x\in [0,\frac{1}{3}] \\ \\ -3x+2, & \text{if } x\in (\frac{1}{3}, \frac{2}{3}] \\ \\ 3x+2, & \...
6
votes
2answers
391 views

How to implement a numerically efficient Airy Zeta Function

Define the Airy zeta function for $n=2,3,\dots$, by $$ Z(n) := \sum_r \frac{1}{r^n}. $$ where the sum is over the zeros $r$ of the Airy function $\operatorname{Ai}$. In Mathematica the $\operatorname{...
5
votes
2answers
273 views

Only perform a symbolic differentiation once

I want to define a function that involves a differentiation step that Mathematica can do easily, which might be of the form ...
1
vote
2answers
252 views

Function to Represent Recursive Integral

I'd like to represent the following recursive integral equation to evaluate/graph (for $n\leq3$): $$K_{i,n}\left(x\right)=\int_{x}^{\infty}K_{i,n-1}\left(y\right)dy$$ where $$K_{i,0}\left(x\right)=...
0
votes
2answers
231 views

Super recursive function

I have read about how to define a recursive function using RecurrenceTable but for $u(100)$ you need all 99 previous terms. The posts doesn't help me. I want to ...
9
votes
1answer
1k views

Compiling a recursive formula

My question is related to computing what is called "invariant measure" for a particular well known fractal - the Sierpinski triangle. We have an array m of four two by two matrices, say ...
3
votes
1answer
83 views

Functions that remember some arguments while not remembering other arguments

I would like to some programming that is very generic. Particularly I am interested in the following: Let's say I want to write a function ...
3
votes
1answer
234 views

Recursion with substitution mathematica

When writing out some mathematical equation like we see on the picture here I got the problem of writing out all terms. I know what recursion means and have done recursion for example for the ...
2
votes
1answer
152 views

Numerical evaluation of ChebyshevT

When I evaluate the following Chebyshev series of the first kind in two different ways, I get two very different results: ...
0
votes
1answer
68 views

Memoization (or otherwise) for a recurrent definition of a (double) sequence of functions

Beside I've found answers to similar questions (e.g. this), I can't adapt them to my case. I am interested in the following sequence of functions $p_{m,n}(t)$, where $m$ and $n$ are integer: $$ \...
0
votes
1answer
120 views

Create recursive sequence of functions with memoization

I am trying to create a sequence of functions and have it properly memoize the results. The recursive operation is simply convolution, so it possible there is a better way to do this (obviously, if I ...
0
votes
1answer
140 views

Nested integration

I wish to perform the following nested integral: \begin{align} I_n=\int_{-\infty}^\infty dx_n~f(x_n,x_{n+1})\int_{-\infty}^\infty dx_{n-1}~f(x_{n-1},x_n)...\int_{-\infty}^\infty dx_1~f(x_1,x_2)\int_{-\...
0
votes
1answer
412 views

Tabulating Numeric Approximation

I was wondering how to approximate or tabulate values for this numeric approximation: It is the following: The confusing part is how to implement the subscripts in mathematica. $y_{i+1} = (t_i - y_i)...
43
votes
0answers
2k views

Fast Spherical Harmonics radiative transfer

This is a rather specific question and I apologize for spamming you with some lengthy code. But it could be interesting for some reader and maybe you can help out, so please bear with me. I am using ...
0
votes
0answers
82 views

Dummy variable in nested integral operator

I am trying to make a nested integral operator where the output function of one step must be applied to the dummy variable to be integrated in the next step. Here my code: ...
0
votes
0answers
70 views

Reuse symbolic polynomial

Suppose I have a function that returns a polynomial in x, GetPoly[params_,x_]:=GetPoly[params,x]... but it is quite slow, so ...
0
votes
0answers
43 views

Keeping memory to reduce the running time of recursion [duplicate]

I am using the following recursion in Mathematica to compute W[n, 1, s] for given n and s: ...