One systematic way of accelerating calculations in Mathematica is defining functions as arrays in the following way: suppose a relevant function is f(x), defined in Mathematica as f[x_]:= .... Acceleration is generally achieved by defining f(x) as f[x_]:= f[x] = ... (for instance, if the function at issue is sin(x), one defines it as follows: f[x_]:= f[x] = Sin[x], instead of f[x_]:= Sin[x]). In this way, every time f[x] is called for the same argument, Mathematica does not evaluate it anew, but retrieves the earlier-calculated value from the memory.

In defining functions (at least the most time-consuming ones that are called repeatedly) in Mathematica in the above way, two aspects should be kept in mind: first, Mathematica's use of RAM becomes an increasing function of time, so much so that for certain programs that have a long running time, it may exhaust the RAM of one's machine,* leading to the machine getting out of control, only to be revived by brute-force rebooting. Therefore, in running programs written in the above way, one has to keep a close eye on the use of RAM by one's machine. Second, for the reliability of the computations, it becomes almost compulsory to quit the kernel after each run. This both clears the RAM and, importantly, prevents possible computational errors from propagating. This propagation of errors has to do with the fact that by calling f(x) for the same value of x as previously, does not lead to a new calculation, but to reading the contents of a memory location. Often, when a program has not terminated naturally in an earlier run, it is likely that a re-run of it, without quiting the active kernel, leads to totally nonsensical outcomes.

As a general rule, I only turn functions into arrays (in the way described above) only after having tested my Mathematica programs. Following this, on turning the functions into arrays, I check whether my original test results are reproduced.

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* To appreciate this, note that defining f[x1_, x2_, ..., xn_] as f[x1_,x2_,...,xn_]:= f[x1,x2,...,xn] = ... is equivalent to defining an n-dimensional array in, say, Fortran.

Dear Behnam,

it is very interesting idea how to speed up calculations in Mathematica. I have to try it in my future calculations. Thanks a lot for a smart clue.

From the other side I think how to define complicated function which uses array (as one variable) to compute series values and gives array as a result  (but not in formal loop as it is used in traditional language programming). In a very simple case

X={x1,x2,….,xn}

f[x_]:=Sin[x];

..f[X];

Do you have any practice in this matter?

Dear Radoslaw, you are welcome. As for your question, no I have as yet not exploited the possibility that you indicate. Perhaps I should mention that I have greatly benefited from the rather concise book Introduction to Mathematica® for Physicists, by Andrey Grozin (Springer, 2014), the link to which I attach below. The book very concisely and intelligibly describes how Mathematica works on patterns (his illustrative examples are excellent); patterns and substitutions are relevant to your question. Incidentally, already the history of computer algebra as described by Grozin in this book is very valuable; he writes as a person who has been directly involved in the area so that his description is both vivid and authoritative.

http://link.springer.com/book/10.1007%2F978-3-319-00894-3

Dear Behnam,

thank  you for your information about valuable book. I found some very good opinion on the Internet about it. I have access to Springer books but the latest are not free for me. I looked through a few pages of this book on the Amazon.

Could you recommend me other books from Springer?

Dear Waldemar,

I completely agree with you that C and Fortran are much more faster than Mathematica. But Mathematica is more user-friendly because of its library.  Some years ago I used Mathematica only as a tool for verifications results which I obtained from my programs written for example in C. Now I think I am “addicted” to Mathematica -  I try to find  new ways to accelerate it. Thanks to Gambit (Cracow and colleagues from Wolfram) I could test Mathematica computing cluster in my laboratory  during a few months.

I appreciate people from AGH and thanks for sending them my question.

Dear Radoslaw, I know two other books on Mathematica published by Springer; one is by Gerd Baumann, Mathematica for Theoretical Physics: Classical Physics and Nonlinear Dynamics (2005), which frankly does not appeal to me (but this is certainly a personal view which may not be shared by others), and one by Michael Trott, in, if my memory is not failing, 3 volumes; I possess only the volume with the title The Mathematica GuideBook for Numerics. Trott's books are in my view mostly reference books that are far too extensive for physicists (the volume I possess counts nearly 1300 pages), except those physicists whose main focus is on programming, to be contrasted with solving physics problems and using programming only as a means to a goal in physics. If you wish to look further afield, that is further than Springer, I would recommend you the book Programming with Mathematica® - An Introduction (Cambridge University Press, 2013) by Paul Wellin -- link attached below. The exercise manual of this book, which is very useful, is in public domain.

Incidentally, there are also some very good public websites on the Internet dedicated wholly to Mathematica. One such website that I have consulted in the past is Mathematica Stack Exchange, the link to which I also attach below.

http://www.cambridge.org/gb/academic/subjects/computer-science/scientific-computing-scientific-software/programming-mathematica-introduction?format=HB

http://mathematica.stackexchange.com/

Dear Behnam,

thank  you for your information about interesting books with your valuable opinion. The first book by Gerd Baumann I will view in a free time. The second one (3 or 4  volumes?) by Trott I had in the past. It is too large and it is rather reference book as you also said. I suppose the third by Wellin may be very useful.   

Thanks for information about good public websites on the internet dedicated Mathematica. As a curiosity I give you the link to interesting e-book which I found a few months ago (may be you do not have this book) written by Shifrin:

http://www.mathprogramming-intro.org/

You are welcome Radoslaw. As it happens I have also the book by Leonid Shifrin. In the course of the past year I have gone through almost everything written on Mathematica (not that I know them all in any detail), but thank you very much for kindly introducing it to me. None of the books I have consulted thus far rivals the book by Grozin, in terms of conciseness and relevance to what I think one should know when using Mathematica. Grozin's book consists of two parts and if you leave out the second part, what remains is extremely concise and to the point. 

Dear Behnam & Dear Radoslaw ,

Thanks you both for your informative conversations.It helped me a lot.