What is the Simpson method and how to implement it in Pascal

To compute the value of the integral, though approximate, there is a beautiful method, named after its Creator, - Simpson method. It is also called the method of parabolas, because it is a parabola. This figure is constructed as close as possible to the function. In fact, so as to construct a parabola, whose points coincide exactly with the points of the function, is not possible, the integral is approximately. The formula for finding its boundaries a and b looks like this: 1/h*(y₀+4y₁+2y₂+4y₃+…+4y_N-1+y_N). Here we just need to calculate each y from 0 to n, where n we define ourselves – the more, the better, because the more y-s, the more approximate to the true value we get. As for h, then this step and is calculated by the following formula: (b-a)/(n-1).

In theory it's pretty simple, but would need to implement all this in practice. For many programmers there is no better way to solve this problem, as the method of Simpson – Pascal or Delphi. In this environment it is easy to calculate the integral, and to construct to it the graph of the function, and even built her a trapezoid. So, let's look at how you can quickly implement Simpson method and, if desired, even to explain how here and that organized, to all interested parties.

But before that, remember how it looks integral. This figure, which is bounded by lines starting at the axis x, that is, a and b.

So, to start the program is to create a function for integrable functions (pardon the tautology), in which you just have to write f:= and we will find the integral. It is extremely important not to make mistakes in entering functions in Pascal. But that is a separate topic of conversation. The resulting code will look something like this:

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Function f(x:real):real;

Main text

Begin

F:=25*ln(x)+sin(10); {here you need to write the contents of your function}

End;

Then write a function to implement Simpson method. It all started like this:

Simpsonmetod function(a,b:real;n:integer):real;

Next, declare the variables:

Var

S:real; { Subtotal (continue) }

H:real; { Step }

My:integer; { Just a counter }

Mno:integer; { Regular multipliers }

And now, in fact, the program itself:

Begin

H:=(b-a)/(n-1); { calculate the step according to the standard formula. Sometimes in the job step is written in this case this formula is not applied }

S:= f(b)+ f(a); { Sets the initial value of the step }

Mno:=4; { Remember the formula 1/h*(y₀+4y_1… here is the 4 here it is written, the second multiplier will be 2, but more on that next }

Now the same basic formula:

For my:=1 to n-2 do begin

S:=s+mno*f(a+h*mu); { add amount To the next multiplier, multiplied by 4*y_N or 2* y_N }

If (mno=4) then mno:=2 else mno:=4;{ Here and change the multiplier if it is now equal to 4, then changed to 2 and Vice versa}

End;

Simpsonmetod:=s*h/3; { Then the resulting cycle is the sum multiplied by h/3 according to the formula}

End.

That was it – do all actions in accordance with the formula. If you still do not understand how to apply in the main program Simpson method, an example will help you with this.

So after writing all write functions

Begin

N:=3; { Set n }

Q:=simpsonmetod(a,b,n); { since the Simpson method is to calculate the integral from a to b, there will be several calculation steps, so make a loop }

Repeat

Q2:=q; { Remember previous step }

N:=n+2;

Q:=simpsonmetod(a,b,n); { calculated value }

Until (abs(q-q2)<0.001);{ written task accuracy, and therefore not yet reached the required accuracy, it is necessary to repeat the same steps }

Here's a he-Simpson method. In fact, nothing complicated, all written very quickly! Now open your Turbo Pascal and start writing the program.