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Consider a two-variable function as follows $$f(a,b)=\frac {T(a,b)+C(a,b)}{V(a,b)} \tag{1}$$ I define a $step$ and then define two new functions using (1) and step as $$f_+(a+step,b)=\frac {T(a+step,b)+C(a+step,b)}{V(a+step,b)} \tag{2}$$ and $$f_-(a-step,b)=\frac {T(a-step,b)+C(a-step,b)}{V(a-step,b)} \tag{3}$$ In fact $f_+$ and $f_-$ are two functions which have been changed by $+step$ and $-step$ for each parameter namely generally a $f_+$ defined as $$f_+(... a_i ...)=f(... a_i+step ...) \tag{4}$$ equations 1-3 were an example. How can I write a general form for $f_+$ and $f_-$ which adds to or subtracts step from a special variable each time and do this for all variables, do I need a loop? For the sake of simplicity suppose function (1).

Example:

I need to step is added to $i$th variable where i is the loop number (in fact I'm running this command inside a outer loop), so if currently loop number is 1, step must be added to $a$ in eq(1) and in the next time when $i$ is 2, step must be added to $b$ in eq(1) and so on. Equation (1) is a special case, but an arbitrary function can have n variable so I need each time one of them (regarding to loop number) changes.

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  • $\begingroup$ The required syntax is almost the same as the traditional math notation: Clear[f, a, b]; f[a_, b_] = (T[a, b] + cc[a, b])/V[a, b]; fplus[a_, b_] = f[a + step, b] Why do you think you need a loop? $\endgroup$ – xzczd Nov 29 '19 at 6:34
  • $\begingroup$ because I need to use these $f_+$ and $f_-$ functions into a loop and in an algorithm where these functions are constructed regarding to number of loop, for example if the loop number is 3, I need to construct a $f_+$ function in which step added to its third variable $\endgroup$ – Wisdom Nov 29 '19 at 6:38
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    $\begingroup$ The order isn't important, right? Then you don't need loop: f @@@ Transpose[IdentityMatrix[3] step + {a, b, c}] $\endgroup$ – xzczd Nov 29 '19 at 6:45
  • $\begingroup$ No, order is important, in fact I don't determine that step must be added to which variable, but the loop number does. So if i be the loop number, step must added to ith variable and the $f_+$ is constructed for that special i $\endgroup$ – Wisdom Nov 29 '19 at 6:50
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    $\begingroup$ It would be nice if you present a minimal example. $\endgroup$ – yarchik Nov 29 '19 at 6:50
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ClearAll[fpm];
fpm[g_, a__][step_, pos_: 1] :=  Module[{arg = MapAt[# + step &, {a}, {pos}]}, g @@ arg]

Examples:

ClearAll[f,t, v, a, b, c];
f[a_, b_] := (t[a, b] + c[a, b])/v[a, b]

fpm[f, a, b][step]

(c[a + step, b] + t[a + step, b])/v[a + step, b]

fpm[f, x, y][-step]

(c[-step + x, y] + t[-step + x, y])/v[-step + x, y]

fpm[f, u, t][-step, 2]

(c[u, -step + t] + t[u, -step + t])/v[u, -step + t]

h = Total[{##}^2] &;
fpm[h, x1, x2, x3, x4, x5, x6][-step, 3]

x1^2 + x2^2 + (-step + x3)^2 + x4^2 + x5^2 + x6^2

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  • $\begingroup$ Thanks, however as I said in the above comment, I need a I need to use these $f_+$ and $f_−$ functions into a loop and in an algorithm where these functions are constructed regarding to number of loop. I think that I can set t in your code equal to my loop number to obtain desire result, right? $\endgroup$ – Wisdom Nov 29 '19 at 6:43
  • $\begingroup$ @Wisdom, not sure how you intend to use loop. Maybe you can use the argument pos as the "loop number" to iterate over the positions as in Table[fpm[f, a, b][step, i], {i, 2}] and Table[fpm[f, a, b][-step, i], {i, 2}]. $\endgroup$ – kglr Nov 29 '19 at 6:57
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    $\begingroup$ I added an example. In fact I wanted to construct $f_+$ and $f_-$ functions inside my loop, but seems your idea is good so that I construct a table of all possible functions and call one of them (regarding the loop number) inside my loop each time, right? $\endgroup$ – Wisdom Nov 29 '19 at 7:04

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