Approximation of Functions

Let f(x)=invertedCOS(x) for EQUATION1 (the principal branch of EQUATION2) Find the polynomial of degree two EQUATION3 which minimizes EQUATION4. *(Please see attachment for all equations)

Numerical Integration

Let P2(x) be the quadratic polynomial interpolating f(x) at x= ... Use a Taylor series expansion of f(x) to show (FORMULA) (SEE ATTACHMENT FOR COMPLETE PROBLEM)

Numerical Integration - Two-point Gaussian Quadrature

Derive the two-point Gaussian quadrature formula for: I(f) = integration from 0 to 1 of f(x)log(1/x)dx (See attachment for full question)

Numerical Integration - Two-point Gaussian Quadrature

Derive the one and two point Gaussian quadrature formula for... with weight function w(x)=x

showing matrix norm

The frobenius norm (which I know is not a natural norm)is defined for an n x n matrix A by ||A||_f = (sum i=1 to n, sum j=1 to n, |a_ij|^2)^1/2 Please show that ||.||_f is a matrix norm. That is, satisfy the five axioms. NOte: _ is subscript

Proof please

Prove that ||x^(k) - x|| <= (||T||^k)(||x^(0) - x||) and ||x^(k) - x|| <= (||T||^k/(1-||T||))(||x^(1)-x^(0)||), where T is an n x n matrix with ||T|| < 1 and x^(k)=Tx^(k-1)+c, k=1,2,..., with x^(0) arbitrary, c belonging to R^n, and x=Tx+c

test

only problems #3 &4-a,(without using any software).

Math - Knots

Only solve 4 part A, and 5 using MATLAB Codes.

newtons method

Please show that when n=1, Newtons method given by: x^k=x^(k-1)-(J(x^(k-1))^-1)(F(x^(k-1)) for k>=1 reduces to the familiar Newton's method given by: P_n=P_n-1 - f(p_n-1)/f'(P_n-1) for n>=1 Note: ^-1 is inverse J is the jacobian matrix The top equation is called newton's method for non linear systems. x is a vecto ...continues

MATLAB \ Least Squares -- solving overdeterminded or inexactly specified of equations in approximation

The solution can ONLY be accepted in Matlab. Matlab 7.0 is preferable. The problem is in the attachment file. Thank you.