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Simpson's rule

Simpson's rule

Simpson's rule. A method for finding approximate values

for specific integrals.

The method starts by dividing the interval on which the integral is defined into equal-width bars connected at the top by quadratic curves that closely approximate the function under investigation.

y = f(x)

The value of fba(x)dx given by Simpson's rule with 2n bars of width h is:

1/3 h[(Yo+Y2n) + 3 4(y1+У3+...) + 2(Y2+Y4+...)]

For example: using the rule with four bars of unit width.

simultaneous equations
simultaneous fpl equations

simultaneous equations. A system consisting of many equations with many unknowns and often linear.

For example:

3x + y = 17
2x - 3y = 12
or
x2y + y2 = 26
x + y = 6

3p + 2q + 5r = 6
or 2p + 3q + 4r = 7
4p + 3q - 2r = 8

Linear simultaneous equations can be solved in various ways:

A) By substitution.

x+3y=11
5x-2y=4

From the first equation we find that y3 -11 = x.
Substituting in the second equation we get:

5(11-3y)-2y = 4
55-15y2y= 4
17y = 51
y = 3
i.e.
x = 11-(3×3)
x=2

Thus (33) - 11 =x
x=2
b) Graphically.

x+3y=11
5x-2y=4

We draw the graphs of the two relationships.

x + 3y = 11
5x-2y = 4

The point of intersection of the two lines (2.3) gives the solution to the equation.

c) By adding and subtracting equations

x+3y=11
5x-2y=4

Multiply the first equation by 5:

5x+15y=55

Then subtract the second equation from it:
17y=51
y=3

Now substitute to find x:

x+3×3=11
x=2

d) By matrices
The two equations can be written:

x+3y=11
5x-2y=4
in the following form:
By applying the inverse matrix:
to both sides of this matrix equation, we get:

Simultaneous equations arise from a number of different physical situations, and the study of their solution formulas constitutes an important branch of algebra. The idea of ​​the matrix was developed through the study of equations of this type.