Notes

Numerical integration
--Quadrature: formula for approximating definite integral
--Derivation is based on taking integral of an n-degree Lagrange polynomial.
--Taking integral of degree 1 Lagrange polynomial -> trapezoidal rule
--Taking integral of degree 2 Lagrange polynomial-> Simpson's rule
--Taking integral of degree 3 Lagrange polynomial->3-point gaussian quadrature.
*practice constructing Lagrange interpolating polynomial.
-Deriving trapezoidal rule from taking degree 1 Lagrange polynomial Ch15 pg 443 in the book
-Newton cotes formula: instances of trapezoidal & Simpson's rule. Formula based on interpolating polynomial at equidistance abscissae. If the endpoints are included, then it's called closed formula. And, if the endpoints are not included, then it's called open formula. Midpoint rule is an example of open Newton cotes formula.
-Error in polynomial interpolation at n+1 points is given by pg443
-Quadrature error is the integral of the error formula in above line
Degree of precision = the degree at which the error of = 0 for all polynomial Pn(x) where n <= p
-Composite quadrature improves the degree of accuracy by dividing the interval of integration into subinterval( error is the sum of error at each subinterval). Error in composite Simpson's method is O(h^4)
Any basic rule using n +1 points has precision of at least n (basic degree of precision)
-Gaussian quadrature: choose particular points(instead of equidistant) of approximation to maximize the degree of precision resulting in family of highest precision method. The n+1 points chosen are the zeros/roots of the n+1 degree Legendre polynomial. Precision of n point gaussian quadrature is 2n+1