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# Guass-Legendre 2-point formula

Posted By: Axelle Schmidt     Category: C Programming     Views: 4400

### Guass-Legendre 2-point formula

The given form in the above equation is I = A0f(x1) + A1f(x2) ----------- (1)

where A0, A1 are the unknown co-efficient. Also x1 and x2 are not fixed at the end points.
Thus we have total four unknowns. to evaluate them we require a system of four equations.
The four conditions can be obtained by taking f(x) = 1, x ,x2 and x3 respectively. And then solving the system. The four equations can be given as below:
A0f(x1) + A1f(x2) =  ---------------------- (2)
A0f(x1) + A1f(x2) =  ---------------------- (3)
A0f(x1) + A1f(x2) =  ---------------------- (4)
A0f(x1) + A1f(x2) =  ---------------------- (5)

Placing the value of integration in above equations we get:

A0 + A1 = 2         ------------------ (6)
A0x1 + A1x2 = 0 ------------------ (7)
A0x12 + A1x22 = 2/3 ------------------ (8)
A0 x13+ A1x23 = 0 ------------------ (9)

Dividing  equation 9 by x1 we get A0 x12+ A1x23/x1 = 0 ------------------ (10)
adding equation 10 and 8 we get:

A0x12 + A1x22 = 2/3
-A0x12 + A1x23/x1 = 0
A1x22 - A1x23/x1 = 2/3 ---------------------------- (11)

From equation 7 we have A0 = -A1 (x2/x1) replacing this in equation 11 we get:

A1x22 - A0x22 = 2/3 ----------------------------(12)
=> x22(A0 + A1) = 2/3 => 2x22 = 2/3 => x2 = 1/sqrt(3)

Now dividing equation 8 by x1 we get A0x1 + A1x22/x1 = (2/3) x1 -------------------(13)
subtracting equation 7 from 13 we get:

A0x1 + A1x22/x1 = (2/3) x1
-A0x1 - A1x2 = 0
A1(x22/x1) - A1x2 = (2/3) x1 --------------------- (14)

replacing replacing A1(x2/x1) in equation 11 we get:

-A0x2 -A1x2 = (2/3)x1 => x2(A0 + A1) = -(2/3)x1=> x1 = -(1/sqrt(3))

Now placing value of x1 and x2 in equation 7 we get: -A0+ A1 = 0  => A0 = A1
adding above equation to 6 we get

A0 + A1 = 2
-A0 + A1 = 0
2A1 = 2                          =>  A1 = 1 and A0 = 1

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 Axelle Schmidt author of Guass-Legendre 2-point formula is from Frankfurt, Germany. View All Articles

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