In engineering computations, use of Chebyshev's formula of approximate integration is frequently made.
Let it be required to compute .
Replace the integrand by the Lagrange interpolation polynominal and take certain values of the function on the interval where are any points of the interval
We get the following approximate formula of integration:
After some computaton, it takes the form
where the coefficients are calculated by the formula
Formula (3) is cumbersome and inconvenient for computation because the coefficients are expressed by complex fractions.
Chebyshev posed the inverse problem: specify not the abscissae but the coefficients and determine the abscissae .
The coefficients are specified so that formula (3) should be as simple as possible for computation. This will occur when all coefficients are equal:
If we denote the total value of the coefficients by , formula (3) will take the form
Formula (5) is, generally speaking, an approximate equation, but if is a polynomial of degree not higher than then the equation will be exact. This circumstance is what permits determining the quantities .
To obtain a formula that is convenient for any interval of integration, let us transform the interval of integration into the interval To do this, put
then for we will have and for
where denotes the function of t under the integral sign. Thus, the problem of integrating the given function on the interval can always be reduced to integrating some other function on the interval
To summarize, the problem has reduced to choosing in the formula
the numbers so that this formula will be exact for any function of the form
It will be noted that
On the other hand, the sum on te right side of (6) will, on the basis of (7), be equal to
Equating expressions (8) and (9), we get an equation that should hold for any
Equate the coefficients of on the left and right sides of the equation:
From these equations, we find the abscissas . These solutions were found by Chebyshev for various values of The following solutions are those that he found for cases when the number of intermediate points is equal to 3, 4, 5, 6, 7, 9:
Thus, on the interval an integral can be approximated by the following Chebyshev formula:
where is one of the numbers 3, 4, 5, 6, 7, 9, and are the numbers given in the table. Here, cannot be 8 or any number exceeding 9, for then the system of equations (10) yields imaginary roots.
When the given integral has limits of integration and the Chebyshev formula takes the form
where for and have the values given in the table.
First, by a change of variable, transform this integral into a new one with limits of integration -1 to 1:
Compute the latter integral by Chebyshev's formula, taking