Ну просто есть такие функции которые являются производными функций не выражающимися через элементарные. Первообразная заданной тобой функции является такой, и через элементарную функцию не выражается, поэтому точное значение определить нельзя. Вольфрам в пределах интегрирования выдал мне: 2.66142, но это лишь приближенная аппроксимация. Аппроксимация с точностью до 4 знака после запятой метода Симпсона (в Паскале) выдала 2.5968...
А не пардон, мой косяк, sqr как sqrt прочел. Скину через 20 минут код.
Добавлено через 13 минут
FPC Версия
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| {$mode objfpc}{$H+}
uses crt;
const a=1;
b=4;
eps=0.0001;
type Tfunc=function(arg:real):real;
var F,G,H,P,L:Tfunc;
n:integer;
method,epskoef:byte;
rkey:char;
function pow(x:real; y:byte):real;
var i:byte;
b:real;
begin
b:=1;
for i:=1 to y do
b:=b*x;
pow:=b;
end;
{$F+}
function IntegratingFunc(arg:real):real;
begin
IntegratingFunc:=sqr(ln(arg));
end;
function AntiDeriv(arg:real):real;
begin
AntiDeriv:=arg*(sqr(ln(arg))-2*ln(arg)+2);
end;
function FirstDer(arg:real):real;
begin
FirstDer:=2*ln(arg)/arg;
end;
function SecDerivative(arg:real):real;
begin
SecDerivative:=(2-2*ln(arg))/sqr(arg);
end;
function FourthDer(arg:real):real;
begin
FourthDer:=(22-12*ln(arg))/pow(arg,4);
end;
{$F-}
function Shag(num:integer):real;
begin
Shag:=(b-a)/num;
end;
function Residue(arg:Tfunc; num:real; meth:byte):real;
begin
case meth of
1:
Residue:=-(((sqr(num)*(b-a))/12)*arg((b-a)/2));
2:
Residue:=-(((pow(num,4)*(b-a))/180)*arg((b-a)/2));
3:
Residue:=sqr(num)*(b-a)*arg((b-a)/2)/24;
4,5:
Residue:=((num*(b-a))/2)*arg((b-a)/2);
end;
end;
function ValueofX(ind:integer; num:integer):real;
begin
ValueofX:=a+ind*Shag(num);
end;
function SumOne(arg:Tfunc; num:integer):real;
var i:integer;
sum:real;
begin
sum:=0;
for i:=1 to num-1 do
sum:=sum+arg(ValueofX(i,num));
SumOne:=sum;
end;
function SumTwo(arg:Tfunc; num:integer):real;
var i:integer;
sum:real;
begin
sum:=0;
for i:=1 to num-1 do
if (i mod 2 =0) then
sum:=sum+2*arg(ValueofX(i,num))
else sum:=sum+4*arg(ValueofX(i,num));
SumTwo:=sum;
end;
function SumThree(arg:Tfunc; num:integer; meth:byte):real;
var i:integer;
sum:real;
begin
sum:=0;
case meth of
3:
for i:=0 to num-1 do
sum:=sum+arg(ValueofX(i,num)+Shag(num)/2);
4:
for i:=0 to num-1 do
sum:=sum+arg(ValueofX(i,num));
5:
for i:=1 to num do
sum:=sum+arg(ValueofX(i,num));
end;
SumThree:=sum;
end;
function ResultIntegrate(arg1,arg2:Tfunc; num:integer; meth:byte):real;
begin
case meth of
1:
ResultIntegrate:=Shag(num)*(((arg1(ValueofX(0,num))+arg1(ValueofX(num,num)))/2)+SumOne(arg1,num)+Residue(arg2,Shag(num),meth));
2:
ResultIntegrate:=Shag(num)*(arg1(ValueofX(0,num))+arg1(ValueofX(num,num))+SumTwo(arg1,num)+Residue(arg2,Shag(num),meth))/3;
3,4,5:
ResultIntegrate:=Shag(num)*SumThree(arg1,num,meth)+Residue(arg2,Shag(num),meth);
end;
end;
function NewtonLeib(arg:Tfunc):real;
begin
NewtonLeib:=arg(b)-arg(a);
end;
function Percent(arg1,arg2:real):real;
begin
if arg2>arg1 then
Percent:=(arg1/arg2)*100
else
Percent:=(arg2/arg1)*100;
end;
begin
clrscr;
F:=@IntegratingFunc;
G:=@SecDerivative;
H:=@AntiDeriv;
P:=@FourthDer;
L:=@FirstDer;
writeln('Choose method of approximation: ');
writeln('Trapezodial method-Press "1";');
writeln('Simpson method-Press "2";');
writeln('Medium Rectangle method-Press "3";');
writeln('Left Rectangle method-Press "4";');
writeln('Right Rectangle method-Press "5"');
rkey:=readkey;
case rkey of
'1': begin
n:=40;
method:=1;
epskoef:=3;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,G,n,method)-ResultIntegrate(F,G,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,G,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,G,2*n,method), NewtonLeib(H)):1:8);
end;
'2': begin
n:=40;
method:=2;
epskoef:=16;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,P,n,method)-ResultIntegrate(F,P,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,P,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,P,2*n,method), NewtonLeib(H)):1:8);
end;
'3': begin
n:=40;
method:=3;
epskoef:=3;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,G,n,method)-ResultIntegrate(F,G,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,G,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,G,2*n,method), NewtonLeib(H)):1:8);
end;
'4': begin
n:=40;
method:=4;
epskoef:=3;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,L,n,method)-ResultIntegrate(F,L,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,L,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,L,2*n,method), NewtonLeib(H)):1:8);
end;
'5': begin
n:=40;
method:=5;
epskoef:=3;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,L,n,method)-ResultIntegrate(F,L,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,L,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,L,2*n,method), NewtonLeib(H)):1:8);
end;
end;
readln;
end. |
|
ABC Версия
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| uses crt;
const a=1;
b=4;
eps=0.0001;
type Tfunc=function(arg:real):real;
var F,G,H,P,L:Tfunc;
n:integer;
method,epskoef:byte;
rkey:char;
function pow(x:real; y:byte):real;
var i:byte;
b:real;
begin
b:=1;
for i:=1 to y do
b:=b*x;
pow:=b;
end;
{$F+}
function IntegratingFunc(arg:real):real;
begin
IntegratingFunc:=sqr(ln(arg));
end;
function AntiDeriv(arg:real):real;
begin
AntiDeriv:=arg*(sqr(ln(arg))-2*ln(arg)+2);
end;
function FirstDer(arg:real):real;
begin
FirstDer:=2*ln(arg)/arg;
end;
function SecDerivative(arg:real):real;
begin
SecDerivative:=(2-2*ln(arg))/sqr(arg);
end;
function FourthDer(arg:real):real;
begin
FourthDer:=(22-12*ln(arg))/pow(arg,4);
end;
{$F-}
function Shag(num:integer):real;
begin
Shag:=(b-a)/num;
end;
function Residue(arg:Tfunc; num:real; meth:byte):real;
begin
case meth of
1:
Residue:=-(((sqr(num)*(b-a))/12)*arg((b-a)/2));
2:
Residue:=-(((pow(num,4)*(b-a))/180)*arg((b-a)/2));
3:
Residue:=sqr(num)*(b-a)*arg((b-a)/2)/24;
4,5:
Residue:=((num*(b-a))/2)*arg((b-a)/2);
end;
end;
function ValueofX(ind:integer; num:integer):real;
begin
ValueofX:=a+ind*Shag(num);
end;
function SumOne(arg:Tfunc; num:integer):real;
var i:integer;
sum:real;
begin
sum:=0;
for i:=1 to num-1 do
sum:=sum+arg(ValueofX(i,num));
SumOne:=sum;
end;
function SumTwo(arg:Tfunc; num:integer):real;
var i:integer;
sum:real;
begin
sum:=0;
for i:=1 to num-1 do
if (i mod 2 =0) then
sum:=sum+2*arg(ValueofX(i,num))
else sum:=sum+4*arg(ValueofX(i,num));
SumTwo:=sum;
end;
function SumThree(arg:Tfunc; num:integer; meth:byte):real;
var i:integer;
sum:real;
begin
sum:=0;
case meth of
3:
for i:=0 to num-1 do
sum:=sum+arg(ValueofX(i,num)+Shag(num)/2);
4:
for i:=0 to num-1 do
sum:=sum+arg(ValueofX(i,num));
5:
for i:=1 to num do
sum:=sum+arg(ValueofX(i,num));
end;
SumThree:=sum;
end;
function ResultIntegrate(arg1,arg2:Tfunc; num:integer; meth:byte):real;
begin
case meth of
1:
ResultIntegrate:=Shag(num)*(((arg1(ValueofX(0,num))+arg1(ValueofX(num,num)))/2)+SumOne(arg1,num)+Residue(arg2,Shag(num),meth));
2:
ResultIntegrate:=Shag(num)*(arg1(ValueofX(0,num))+arg1(ValueofX(num,num))+SumTwo(arg1,num)+Residue(arg2,Shag(num),meth))/3;
3,4,5:
ResultIntegrate:=Shag(num)*SumThree(arg1,num,meth)+Residue(arg2,Shag(num),meth);
end;
end;
function NewtonLeib(arg:Tfunc):real;
begin
NewtonLeib:=arg(b)-arg(a);
end;
function Percent(arg1,arg2:real):real;
begin
if arg2>arg1 then
Percent:=(arg1/arg2)*100
else
Percent:=(arg2/arg1)*100;
end;
begin
clrscr;
F:=IntegratingFunc;
G:=SecDerivative;
H:=AntiDeriv;
P:=FourthDer;
L:=FirstDer;
writeln('Choose method of approximation: ');
writeln('Trapezodial method-Press "1";');
writeln('Simpson method-Press "2";');
writeln('Medium Rectangle method-Press "3";');
writeln('Left Rectangle method-Press "4";');
writeln('Right Rectangle method-Press "5"');
rkey:=readkey;
case rkey of
'1': begin
n:=40;
method:=1;
epskoef:=3;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,G,n,method)-ResultIntegrate(F,G,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,G,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,G,2*n,method), NewtonLeib(H)):1:8);
end;
'2': begin
n:=40;
method:=2;
epskoef:=16;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,P,n,method)-ResultIntegrate(F,P,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,P,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,P,2*n,method), NewtonLeib(H)):1:8);
end;
'3': begin
n:=40;
method:=3;
epskoef:=3;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,G,n,method)-ResultIntegrate(F,G,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,G,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,G,2*n,method), NewtonLeib(H)):1:8);
end;
'4': begin
n:=40;
method:=4;
epskoef:=3;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,L,n,method)-ResultIntegrate(F,L,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,L,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,L,2*n,method), NewtonLeib(H)):1:8);
end;
'5': begin
n:=40;
method:=5;
epskoef:=3;
clrscr;
write('Approximation...');
while ((abs(ResultIntegrate(F,L,n,method)-ResultIntegrate(F,L,2*n,method))/epskoef)>eps) do
n:=2*n;
clrscr;
writeln('Result of integrating: ',ResultIntegrate(F,L,2*n,method):1:8);
writeln('Accurate result: ',NewtonLeib(H):1:8);
writeln('Accuracy: ',Percent(ResultIntegrate(F,L,2*n,method), NewtonLeib(H)):1:8);
end;
end;
readln;
end. |
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Поидее Турбик должен со второй работать. Практически никогда в нем не работал, не знаю как он работает с процедурным типом.
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