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AJL]ЖФФФФФL№ЖЖФФЉ№№№Ф˜ЖФЖФ&№Ф&№ђ№т:ž,ЖЖ&ФИ  у&pdФЪЖ€\Ъ &П0ядR‹\”‹&№ЪЪЖЖЖЖй Numerical integration and differentiation Difference between integration and differentiation Differentiation First order forward f'(x)=[f(x+h)-f(x)]/h First order backward f'(x)=[f(x)-f(x-h)]/h Second order f'(x)=[f(x+h)-f(x-h)]/2h Second derivative (Second order) f''(x)=[f(x+h)+f(x-h)-2f(x)]/h2 Interpolation Interpolate lagrange using two points (x1,y1), (x2,y2) f(x)=(x-x1)y2/(x2-x1)+ (x-x2)y1/(x1-x2) df(x)/dx=(y2-y1)/(x2-x1) Interpolate lagrange using three points (x1,y1), (x2,y2),(x3,y3) f(x)=(x-x1)(x-x2)y3/(x3-x1)(x3-x2)+ (x-x1)(x-x3)y2/(x2-x1)(x2-x3)+ (x-x3)(x-x2)y1/(x1-x3)(x1-x2) df(x)/dx=(2x-x1-x2)y3/(x3-x1)(x3-x2)+ (2x-x1-x3)y2/(x2-x1)(x2-x3)+ (2x-x3-x2)y1/(x1-x3)(x1-x2) Integration We only discuss Riemann integrals Lim S№i=1nf(xi)D№xi Open  without the boundaries Close - with the boundaries Integration formulae - S№i=1n wi f(xi) We can optimize either wi or the choice of xi or both Lagrange integrals  approximate function by lagrange polynomial and integrate the polynomials f(x)=S№f(xi)li(x) (f(x)dx=(S№f(xi)li(x)dx=S№f(xi) (li(x)dx wi=(li(x)dx Method of undetermined coefficients Equivalent request that integral is precise for all polynomials until degree n-1. S№i=1n wi f(xi) =(f(x)dx for all polynomials up to degree n-1 Solution equivalent to Lagrange integration. S№i=1n wi xik =(xkdx=xk+1/(k+1)= [bk+1-ak+1]/(k+1) Degree of integration  precision of integral. If integral is precise for polynomial of degree n-1 then the integration is precise to degree n. Newton quadratures fix xi, select wi Midpoint rule  constant at center M(f)=S№i=1n-1f([xi+xi+1]/2)D№xi Trapezoid rule  linear in region T(f)=S№i=1n-1[f(xi)+f(xi+1)]/2D№xi Simpsons rule  Second order interpolation S(f)=S№i=1n-1[f(xi)+ 4f([xi+xi+1]/2)+f(xi+1)]/6D№xi Estimate errors for quadratures The error for the midpoint rule in a given interval h is f''(c)h3/24 where c is a point in the interval (from taylor expansion in the middle of the section). The error for the trapezoid rule is approximately f''(c)h3/12 (from the sum of talyor expansion on the two sides) Simpson error is f(4)(c)/2880h5 two order higher precision than both trapezoid and midpoint. Gauss quadratures Selact both xi, and wi to optimize order of integration. In interpolary integration, we had n free parameters, we obtained a degree of n-1, if we vary both xi and wi we can obtain a 2n-1 degree Method of undermined coefficients for both xi and wi results in 2n non linear equations. Choose an orthogonal polynomial (legender) of degree 2n-1 that is orthogonal to all polynomials of lower degree. The zeros of this polynomial are real and simple and in the integration interval. 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