## Performance issue with evalf on integral...

I have many integrals which I would like to calculate the value. The one in attachment is the simpliest example.

It shows 'too many level of recursion',

I know that it has something to do with the piecewise, however, it shouldn't, right? Any insights?

evalfandintPerformance.mw

evalfandintPerformance.pdf

## evalf random results - serious bug...

 > restart;
 > Digits:=10; to 10 do evalf(add(sin(k), k = 1 .. 10000)) od;
 (1)
 > restart;   # execute several times to obtain randomness
 > interface(version);
 (2)
 > Digits:=18;
 (3)
 > to 10 do   evalf(add(sin(k), k = 1 .. 10000)) od;
 (4)
 >

## a problem with integral command (evalf(Int))...

when i use the command evalf(Int(f(y),y=-b..b)), i expect that the output of this command to be an integer. but the output is to form of below:

i think that the problem is because of the form of the function f(y) that is the form of below:

f(y) = a*10^354*(b*10^-356*g(sin(y) , cos(y) and exp(y)))

but i dont know how i solve this problem:(

## How to substitute values from multiple lists into ...

Dear All

For six parameters, I have corresponding list of their values and there are eight values for every parameter. I need to put these values in a formula to obtain a list of output values. There are two formulas one for 'P' and next is for 'RL'. I have used value of 'P' to calculate value of 'RL'. There are some complex number too, for which I have used modulus and final value is calculated by using 'evalf', but this command is not returning proper values for list as required. But this command works fine when I use single value from every list to calculate RL.

The Maple sheet attached herewith.

List.mw

Regards

## Evaluate min and max in Excel...

Hello,

a=number      b=number

=maple("Qm:=x->(diff(KelvinBei(0,x),x)*psi2(x)-(diff(KelvinBer(0,x),x)*psi1(x)))/&1";B11)

=maple("Qv:=x->(&1*psi2(x)-(&2*psi1(x)))/(&3*&4)";B6;B7;B2;B11)

=maple("Fm:=x->(Qv(x)+(&1*Qm(x)))/2";B3)

I need abs(max(Fm(x))) and abs(min(Fm(x))) values of function Fm(x), locals, for a<x<b in excel.

Now I use a vector to do this, but I need an exact values not an approximation of a fuction evaluated with n values of x.

what I do:

=maple("seq(i,i=&1..&2,&3)";N2;N3;N4)

=maple("A:=&1";N5)

=maple("G:=map(g->evalf(eval(Fm(x),x=g)),[A])")

=maple("max(abs~(G))")

Someone can help me??

## Why the evaluation of root of a real number is so ...

Dear friends

It seems that Maple takes a long time to evaluate the square roots of numbers.

See the simple code below.

st := time();

for i to 1000 do for j to 1000 do

a[i, j] := evalf(abs(i-j+1)^0.3-abs(i-j)^0.3):

end: end:

time()-st

I run it, then after a few seconds I run it again and again  to see the consuming time: once the running time is 77 seconds, then is 57 seconds, again is 73 seconds ...

Two questions:

1- Why the time is so differnt?

2- Why a simple code is being done at about a minute? Based on the number of operations, I think it should be done at less than a second. It just involves finding two million real third roots each of them less than 100 operations (if Newton method for finding roots is applied it probably needs less than 20 operations). I was thinking that a computer may do one billion operations per second.

Since I need to report my numerical results in a scientific paper, it is important for me to know what's going on.

It is worthy of noting that I use Maple 18 on a Lenovo Laptop with Corei3 1.90 GHz with 64 bit operating system and 4 Gb RAM.

In advance, I appreciate for helping me to reveal the secrets.

Thank you all

## Sums of certain series and a bug...

This question is related to the recent post
http://www.mapleprimes.com/questions/211460-Series-Of-Bessel-Functions

1. Consider the following fast convergent series:

f:=n->(-1)^(n+1)*1/(n+exp(n));
S1:=Sum(f(n),n=1..infinity);
evalf(S1);
S2:=Sum(f(2*n-1)+f(2*n),n=1..infinity);
evalf(S2);

As expected, the sum of the series is obtained very fast (with any precision), same results for S1 and S2.

2. Now change the series to a very slowly convergent one:

f:=n->(-1)^(n+1)/sqrt(n+sqrt(n));

evalf(S1) is computed also extremely fast, because the acceleration algorithm works here perfectly.
But evalf(S2) demonstrates a bug:

Error, (in evalf/Sum1) invalid input: `evalf/Sum/infinite` expects its 2nd argument, ix, to be of type name, but received ...

3. Let us take another series:

f:=n->(-1)^(n+1)/sqrt(n+sqrt(n)*sin(n));

Now evalf(S1) does not evaluate numerically and evalf(S2) ==> same error.
Note that I do not know whether this series is convergent or not, but the same thing happens for the obviously convergent series

f:=n->(-1)^(n+1)/sqrt(n^(11/5)+n^2*sin(n));

(because it converges slowly (but absolutely) and the acceleration fails).
I would be interested to know a method to approximate (in Maple) the sum of such series.

Edit. Now I know that the mentioned series

converges (but note that Leibniz' test cannot be used).

## Inconsistency in Maple 17...

evalf(5^1.25, 30);
7.476743906

evalf(5^(5/4), 30);
7.47674390610610270955949497070

evalf(evalf(5^1.25, 30)-evalf(5^(5/4), 30), 30);
-1.0610270955949497070 *10^(-10)

I would expect the two results to be the same.

I am using Maple 17 and windows 7 operating system.

Is there a reason that  evalf(5^1.25, 30) will not compute 30 decimal places?

## Sequence of values of given funciton...

Dear all,

Thank you for helping me  to generate a table of values of f(x) starting with x=0 to 100 in steps of 1, that is for x=0,1,2,3,...,100.

I tried:

f:=x->2*sqrt(3)*a1*a2*(sum(pochhammer(1/3,k)*3^k*x^(3*k)/(3*k)! ,k=0..infinity)*sum(pochhammer(2/3,k)*3^k*x^(3*k+2)/(3*k+2)!  ,k=0..infinity)-sum(pochhammer(2/3,k)*3^k*x^(3*k+1)/(3*k+1)!  ,k=0..infinity)*sum(pochhammer(1/3,k)*3^k*x^(3*k+1)/(3*k+1)!  ,k=0..infinity));

tab_values:=[evalf(simplify(seq(Ni1(xx),xx=0..100)))];

But I the result is amazing.... I don't understand the problem.

Thanks

## Problem with plot...

Dear all;

I need you to understand this problem...

when i plot the function, using the graph i see that the function is above the x-axis but when I compute some values of this function I get a negative values....like
evalf(y(99.6));  is a negative value, but in the graph it is possible... I don't  undertand the problem...

restart:
with(plots):

# funciton

y:=x->-4.1123583570*10^281*exp(-(2/3)*x^(3/2))/(x^(1/4)*sqrt(Pi))+1.6554662320*10^(-289)*exp((2/3)*x^(3/2))/(x^(1/4)*sqrt(Pi))+(16/153)*x^(7/6)*sqrt(Pi)*exp((2/3)*x^(3/2))+Pi*((1/2)*exp(-(2/3)*x^(3/2))*(-1+exp((2/3)*x^(2/3)))/(x^(1/4)*Pi)-(16/153)*x^(7/6)*exp((2/3)*x^(3/2))/sqrt(Pi)):

#I plot this function in the interval (a,b)

a:=99;b:=100; # interval (a,b)
forget(evalf): Digits:=20:
P1:=plottools:-transform((x,y)->[x+a,y])(plot(expand(y(x+a)),x=0..1,color=blue)):
forget(evalf): Digits:=4000:
P2:=plot(ysol, a..b, style=point, adaptive=false, numpoints=25, symbol=solidcircle, symbolsize=20, color=blue):
Digits:=20:
plots:-display(P1,P2);
evalf(y(99.6));

## Different values of numerical integration...

Dear friend,

Recently I noticed, that numerical integration returns different values for the same function.

For example the code

restart;
evalf(int((exp(x)*(4420*cos(4)*sin(4)-544*cos(4)^2+148147*exp(-1)-4225*cos(4)-215203)/(71825*exp(1)-71825*exp(-1))-exp(-x)*(4420*cos(4)*sin(4)-544*cos(4)^2+148147*exp(1)-4225*cos(4)-215203)/(71825*exp(1)-71825*exp(-1))+(32/4225)*cos(4*x)^2+(1/71825)*(4225+(2210*x-6630)*sin(4*x))*cos(4*x)+x^2+8434/4225)^2, x = 0 .. 1));

each time returns values

0.0005951015934
0.0005950850548
0.0005950974588
0.0005950960805
0.0005951297843 etc.

Maybe, evalf uses a stochastic algorithm for integration?

## Сurse of dimensionality...

I'm trying to solve a system of equations thats expressed as a summation (the original has the summation symbol as opposed to 'sum'):

d_actual := solve(W_actual = sum(W_guess(def-asp_rad_inverse[i], E, asp_rad[i]), i = 1 .. n), def);

When n<5, I get an answer after a few seconds, but when n is higher, the program sits and 'evaluates' forever...I've waited up to 30 min.

Background:

I have a plate with a number of hemispheres on the surface (# of impacted hemispheres given a force = n). Each has a unique radius and they're listed from largest to smallest in 'asp_rad[]'. I have an equation for the deflection of a single hemisphere as a function of Force and material properties that I have rearranged with respect to Force (W_guess(deflection, E, r)). There's an opposing plate that stays parallel to the original plate while pushing down on the asperities with Force W_actual. The total deflection of the opposing plate is the sum of: 1) the difference between the tallest radius and the impacted radius in question (asp_rad_inverse[]), and 2) the deflection of the impacted radius in question.

I'm attempting to solve for the total deflection of the opposing plate via solving for the 'def' in the summation above, but when I run it, the program is not able to compute a solution.

## How to integrate the function...

Dear All

I am trying to integrate a function, however Maple is not giving me the results. I have tried to use int as well as evalf however I am still not getting the results.

I would be grateful if you could please suggest me a way out. I have attached the maple file for your reference.

ThanksExample.mw

## Possible bug in "fsolve"...

As part of a project, I am numerically estimating the roots of many large polynomials. Occasionally, "fsolve" fails with strange errors related to "fsolve/refine2". Searches for these error messages have turned up nothing.

I've inluded the code below that causes the error on Maple 18.02. I apologize for the polynomial in question being so long, it's the shortest example I have. The error it generates is:

Error, (in fsolve/refine2) invalid input: evalf expects its 2nd argument, n, to be of type posint, but received undefined

Is this a bug? Or am I missing some fsolve option to prevent this? Note that it only happens when the "complex" flag is used.

=========

P := x^14-22702264347017701018473605850972699930097274504938699916055555261201515180511538865331807292689345943133521696082918467714371257277276696385067641909170155322906230250853577229812913946663078548646992393337618113886746876557117483839533553328895358682670189394678910311793504505447628428181885141769168591937690303328913335175451328463754619536253583902806843310134957600949886784187209785783810122275010505534415815566439121541947044486358488039865870455952098827525405324562601732796858645293515431747164008309785658410612354201118685855495413079021176507985235094746401708925593687656572387531020719291601076812080687859808747213536777976702071405128537760507468013438105233313663196919816564525291458692028177366393652501832447863872200682143768513389322886600569382594287138458765510827267842205096062437750804878586024353928794905249283675708441066101095406513448522689302522442783437142289641259057413952301148939774149714785/3195755849586795631956816504521213454239300164039404772924331154185577854140658969534719471406093912112781063157828311505891258148739680804289213024862131311540960306206602785748866445362483281617891374949555209869677857419473553982132073059025609434698683760348542259396937054082293168625919023158753310878489047944378154369352523436731294817697449949932655665007647918855300664365159027040571937825740235967492228453331261542499260943085539271304638576578246276634403350307801994081681247214869084246168101721298760198550961832560608341435638093413744839736250679074198753022491225840288065341597851066663786665723409977381822591654466626645542917017628998630902708076612502066607817250779545511895971357711983287763127653752300554550391349040027472903180009282594974618980021621163037989247901106508257414514187962209356325857887950302223210328647697948055097831009797738621154319922212951316644741457327450027692469090867369598*x^13+19088859498864751331345860430721481446264521641744903691362655800372349704990331481604867685549645823662978708926030541236042951546550966879612333115396628902751820387999904934599090760358886795430484312266737008386396041896213971568537362589851823779617430207078749426022658232278071527361264481524611089324107754031784837527081637219350016169914382322455035364613935875393571579561406195287363628553419822536428710010055920488818415526206620047517917895155637033562338042275152771173240104076821411360366799172066699958543868065037999702280159896040588223787643434915579465270491451613199185385049196526456210057933748521047167538262357063585093474544299142560492581751607753970282443057122762426600024763892341448332834018680513343674283251162037067303651651086278409136799357849452879897251530675098741236156640469815784447341282424004221641529187217962536022784563163918511210513153785881467158114512281634789894107114727680109/1065251949862265210652272168173737818079766721346468257641443718061859284713552989844906490468697970704260354385942770501963752716246560268096404341620710437180320102068867595249622148454161093872630458316518403289892619139824517994044024353008536478232894586782847419798979018027431056208639674386251103626163015981459384789784174478910431605899149983310885221669215972951766888121719675680190645941913411989164076151110420514166420314361846423768212858859415425544801116769267331360560415738289694748722700573766253399516987277520202780478546031137914946578750226358066251007497075280096021780532617022221262221907803325793940863884822208881847639005876332876967569358870834022202605750259848503965323785903994429254375884584100184850130449680009157634393336427531658206326673873721012663082633702169419138171395987403118775285962650100741070109549232649351699277003265912873718106640737650438881580485775816675897489696955789866*x^12-127688837609696458957114129756229560761957972259253280819996356067917173759565012901801561924391178368568146719627801670086606489531437386224078360185442651606983719684283163392876990522586784115059551865746707609765679864632874671595399416688286257053075135779925094175440416074968471245768830366824397599424731191899057489251725430472639828977416853808059394673266682604308077331301860791811476274942568803494246399367164616630866928631772760003749091917886558963952047434319195736393271420111064778587861639539510320744497931007588784407172972776901653630399291814617861650330433072614870207218474263898528043868017109168847074788133295715653324601280999334137328493510780499508083274179117783232296907665583279993325725716354393277745170409349317876378784871325009748734263290375761397883657890413900529632709410443413043575189427898559331856967020187201932742096158736566419271039506140015010172468151681141071869870925420155369/639151169917359126391363300904242690847860032807880954584866230837115570828131793906943894281218782422556212631565662301178251629747936160857842604972426262308192061241320557149773289072496656323578274989911041973935571483894710796426414611805121886939736752069708451879387410816458633725183804631750662175697809588875630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fsolve(P,complex);

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