Skip to main content

function construction - Error computing sum of sum of digits



I've defined a function that computes the sum of the base-b digits of n:


DigitSum[n_, b_] := Total[IntegerDigits[n, b]]

Then I defined a function that computes the sum of the base-b digits of all of the integers up to x:


CumDigitSum[x_, b_] := Sum[DigitSum[n, b], {n, 1, x}]

Using these functions, I get


CumDigitSum[1000000,10]=27000001

which is correct. But then for larger inputs I get nonsense like



CumDigitSum[1000001,10]=500011500011

If I work in a different base, the same thing happens: at exactly 1000001, Mathematica begins computing the sum incorrectly. If I bypass my user-defined functions and just write what I mean, I get the correct answer:


Sum[Total[IntegerDigits[n, 10]], {n, 1, 1000001}] = 27000003

Any idea what could be happening here?



Answer



If you wish to compute the correct values using the method you have chosen you could specify Method -> "Procedural" for Sum:


CumDigitSum[x_, b_] := Sum[DigitSum[n, b], {n, 1, x}, Method -> "Procedural"]


CumDigitSum[1000001, 10]


27000003

However, the problem comes form the fact that Sum attempts to speed the calculation by finding a symbolic equivalent. Let's see what your DigitSum returns with symbolic input:


DigitSum[n, b]


b + n


That's clearly not correct generally! Why does this happen? Mathematica functions often work on arbitrary expressions as well as lists, and that is the case with Total. First the IntegerDigits call remain unevaluated:


IntegerDigits[n, b]


IntegerDigits[n, b]

But then Total adds its arguments as though this were {n, b}:


IntegerDigits[n, b] // Total



b + n

To prevent this you can add a Condition or PatternTest as belisarius recommended in a comment above:


DigitSum[n_?NumericQ, b_] := Total[IntegerDigits[n, b]]

This will block Sum form finding a (false) symbolic equivalent thereby forcing it to use a procedural evaluation, even with your original CumDigitSum definition.


For this particular case it is somewhat cleaner to use Tr in place of Total as it will not sum the arguments of an arbitrary expression:


Tr[IntegerDigits[n, b]]



Tr[IntegerDigits[n, b]]

Therefore:


DigitSum[n_, b_] := Tr[IntegerDigits[n, b]]

By the way, it is not recommended to start user Symbol names with capital letters as these can conflict with built-in functions, now or later.


Comments

Post a Comment

Popular posts from this blog

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

What is and isn't a valid variable specification for Manipulate?

I have an expression whose terms have arguments (representing subscripts), like this: myExpr = A[0] + V[1,T] I would like to put it inside a Manipulate to see its value as I move around the parameters. (The goal is eventually to plot it wrt one of the variables inside.) However, Mathematica complains when I set V[1,T] as a manipulated variable: Manipulate[Evaluate[myExpr], {A[0], 0, 1}, {V[1, T], 0, 1}] (*Manipulate::vsform: Manipulate argument {V[1,T],0,1} does not have the correct form for a variable specification. >> *) As a workaround, if I get rid of the symbol T inside the argument, it works fine: Manipulate[ Evaluate[myExpr /. T -> 15], {A[0], 0, 1}, {V[1, 15], 0, 1}] Why this behavior? Can anyone point me to the documentation that says what counts as a valid variable? And is there a way to get Manpiulate to accept an expression with a symbolic argument as a variable? Investigations I've done so far: I tried using variableQ from this answer , but it says V[1...
Post a Comment
Read more