## How to increase the precision of numerical integra...

I want to calculate the following integral numerically with required precision.

First, the functions are defined:

G1:=-0.9445379894;
f:= (x) -> 0.9/abs(x-0.4)^(1/3)+0.1/abs(x-0.6)^(1/2);
U1 := unapply(-exp(-x)*(evalf(Int(f(t)*exp(t), t = 0 .. x))+G1)/2-exp(x)*(evalf(Int(f(t)*exp(-t), t = 0 .. x))+G1)/2, x);
U:= unapply(-exp(x)/2*(evalf(Int(f(t)*exp(-t),t=0..x))+G1)+exp(-x)/2*(evalf(Int(f(t)*exp(t),t=0..x))+G1), x);

Next, I calculate the integral in numerical form:

evalf(Int(U1(x)^2+U(x)^2-2*f(x)*U(x), x=0..1, digits=4, method = _Gquad));

If I specify digits=4, Maple return the answer -0.4291

If I use digits=5 or larger, Maple return someting like this

Is it possible to increase precision of calculation?

## Apparent bug regarding sum of floors...

I have encountered a behavior of Maple that I find hard to explain and I am hoping for help. The command

sum(floor((exp(Pi)-Pi)*n)/3^n,n=0..infinity);

was meant to be an example of "High-precision fraud" as in the 1992 paper of Borwein and Borwein, and indeed it gives 29/2 to within 531 digits. But I am unable to make Maple see this; indeed I get with evalf(%,1000)

14.50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

I find it hard to guess why Maple gets this wrong, actually. The point of the example is that floor((exp(Pi)-Pi)*n)=20*n-1 for many n, but Maple has no problems finding the first failure at n=1112. It must hence be trying something more advanced than just adding up all summands until the tail sum is small enough to satisfy the precision? I guess an alternative approach would indeed be possible since what is being "floored" is relatively simple, but then that seems to be buggy?

Would be very grateful for any assistance!

Best,

Soren Eilers

## Completely different results ...

I'm not really an expert in maple. I'm stuck with a problem that could be related to the precision of the computational engine...

I defined a function which is relatively complicated as follows:

m := proc (x, l, T) options operator, arrow; (exp(l*(x-T)/(1-exp(-l*T)))*exp(l*T/(1-exp(-l*T)))*(1-exp(-l*T))+(2-exp(-l*T)-exp(l*T))*exp(l*(x-T)/(1-exp(-l*T))))/(exp(l*T)-1)-(l*(x-T)*exp(l*(x-T)/(1-exp(-l*T)))-2*exp(-l*T)+4-2*exp(l*T))/(exp(l*T)-1) end proc

whenever i simplify the equation or factor some of the terms, I get completely different results.

As an example, you can define this function as follows

m := proc (x, l, T) options operator, arrow; (exp(x*l/(1-exp(-l*T)))*(1-exp(-l*T))+(2-exp(-l*T)-exp(l*T))*exp(l*(x-T)/(1-exp(-l*T))))/(exp(l*T)-1)-(l*(x-T)*exp(l*(x-T)/(1-exp(-l*T)))-2*exp(-l*T)+4-2*exp(l*T))/(exp(l*T)-1) end proc

By drawing the function, you can see how things get messy.

plot(m(450, (1/250)*x, 250), x = 0 .. 100)

I know the problem occurs as a results of computation round-off. How could I ameliorate that? I have already increase the precision in the option menu, but that didn't help. By the way, how can I be sure that the anwers maple gives me is correct, except using my intuition?

## Cantor staircase in Maple...

How to realize the Cantor staircase function in Maple? Mathematica has it.

I'd like to stress the three desired properties:

(i) The virtual Cantor function can be evaluated to arbitrary numerical precision.

(ii) For certain arguments, this function automatically evaluates to exact values.

(iii) Mathematical function, suitable for both symbolic and numeric manipulation.

## Displaying too many zeros in output...

Hi everybody,

Ever since i updated to maple 2015, the output of calculations display a disproportionate amount of zeros. For instance:

If i write:

2.3*10^(-4)

i get:

0.000230000000000

Is there anyway to make maple NOT display the last 10 zeros? I've tried Tools ->Options ->Precision -> Round screen display to X. But then it rounds all results to that X amount of digits, wich is also anoying. Like, if i set X to be 10, it'll write 3.000000000 instead of just 3.

Hope someone can help me :-)

Anton

## Evalf precision incorrect...

In the following code, the evalf prints -32.16...

restart; S := 8;

sigma := 8/sqrt(2*Pi);

iprec := 151;

evalf(log[2](-(sum(log(round(2^iprec*exp(-j^2/(2*sigma^2)))/2^iprec*exp(-j^2/(2*sigma^2))))*round(2^iprec*exp(-j^2/(2*sigma^2)))/2^iprec, j = 0 .. 7))));

Change the last line to evalf[20](log[2](....)), and re-run *just that line*. It now prints -66.67...

Change the last line to evalf[200](log[2](...)) and re-run *just that line* again. It now prints -151.24... (I have strong evidence to support that the true value is near -151, so I believe this answer.)

Now remove the precision indicator from the command completely and re-run just the last line. It *still* prints -151.24...!

My two questions are: why do I need evalf[200] to get the first three digits of the answer to be correct? and why does setting the evalf precision and then removing it cause the previous precision to persist?

## Broken Summation Notation...

Below is a Maple ws describing the problem. I have some fixed values pr0 and pr1, and a four variations of a function "pdf". The problem is that when I use the function in a sum (either the "sum" function or the sigma notation), the result is different than if I write out the sum explicitly. It's very puzzling as to why they woudl be different in the first place, but even more strange is that the different versions have *different* deltas. Even version 3, with a delta of 10^-100 is too much for my application (moreover the delta gets larger when I sum over more terms).

In the worksheet below, I compare "sum(f(x),x=3..3)" to "f(3)". I expect this should always be identically 0, but that is not true for any of the four versions of my function. (The same is true for "Sum(f(x),x=3..3)".) However, "sum(f(3),x=3..3)" *is* (blessedly) identically "f(3)". What gives?

 >
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Maple doesn't always like subscripted variables. Is that the problem?

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Maybe it's the binary expansion that's the issue

 >

No procedures, no arrays

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 (1)
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 (2)
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 (3)
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 (4)
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## Numerical precision error after subs...

Hello all,

I'm experiencing an aggravating issue when substituting numerical values into a symbolic expression.  I'm using Maple 14.  I believe the issue may be related to floating point precision.  Either way, it's ruining my life.  If anyone has a solution or a work-around, I'd be most grateful.  The following code reproducibly produces the anomolay on my machine:

vx := Vector( 4, symbol=x ):

# A numerator and a denominator. They're equal.
num := (x[1]*x[3]+x[1]*x[4]+x[2]*x[3]+x[2]*x[4])^2:
den := ((x[1]+x[2])^2*(x[3]+x[4])^2):

# Prints true...
evalb(num=den);

# This equals zero
y := 0.1 - 0.1*num/den;

# Prints 0...
simplify(y);

# Prints 0...
simplify(subs(x[3]=1,x[4]=3,0.1-0.1*num/den));

# Prints 0...
simplify(subs(x[3]=1,x[4]=3,y));

# Prints 0.1 x 10^-10 !?!?
simplify(subs(x[3]=1,x[4]=2,y));

## Secant method in double precision...

I need to show what happens to the zero r=20 of f(x)= (x-1)(x-2)..(x-20)-(1/10^8)*(x^19) and the hint given is that the secant method in double precision gives an approximate in [20,21].

At present, I'm calling the secant method on f with a tolerance of 1/(10^12) with an initial x=20, but I'm stuck as to what the second initial value would be. What is the right approach to this question?

## recursive sequence conjecture and proof...

• here is an exercise I got from a text book                                                                                                              calculate the first 10 terms of the following sequence :

u[n+1]=1/2(u[n]+2/u[n]) n>=0

• estimate the differences u[3]-sqrt(2) , u[4]-sqrt(2), u[5]-sqrt(2), and u[6]-sqrt(2) with a precision of 50 numbers
• what can we conjecture about the sequence ?
• how to prove that conjecture with MAPLE ?

## How to save array with smaller precision...

I have saved array calculated with Digits:=5000. Now I want to save it with Digits:=2500. I tried this

Digits := 2500;

LArr2500 := LArr5000;

save LArr2500, "C:\\math\\lambda2500.mpl"

But this don't work...

## How to set precision of output of writedata?...

Hey,

I am using maple to do some numerics (bound to use it instead of mathematica because of university guidelines) and my problem is that as the plotting options of maple are a bit unsatisfactory I need to export my data. This has worked fine so far using the writedata-command while I was working on normal (10 digit) precision. Now however I need to increase the precision and to my dismay writedata just cuts everything after the normal 10 digits away and...

## accuracy of calculation in do loop....

Hello,

I want to calculate the IDs product :