Problem 1 (paper) Using pencil, paper and calculator or Scilab interactively: (a) Write out the explicit form of the Newton-Raphson iteration for the nonlinear system

x²₁ + x²₂ = 0

2x₁x₂ = 1

(b) Starting at x₁=0 , x₂=1 perform one iteration of the method.

Problem 2 (paper) Using pencil, paper and calculator or Scilab interactively: (a) Write out the explicit form of the Newton-Raphson iteration for the nonlinear system

16x⁴ + 16y⁴ + z⁴ = 16

X² + y² + z² = 3

x³ y= 0

(b) Starting at x=y=z=1 perform two iterations of the method.

Problem 3 (paper) Using pencil, paper and calculator, form the Lagrange interpolating polynomial corresponding to the three data points: (0,2),(1,1),(2,1). Express it in standard form . Evaluate this polynomial at x=1.5.

Problem 4 (paper) Calculate the 1st order Newton polynomial that interpolates the two data points (1,1),(2,3). Now extend this to a 2nd order Newton polynomial that in addition interpolates the data point (3,2). (In both cases leave the polynomials in Newton form, for example

c₁ + c₂ (xx₁) + c₃ (xx₁) (xx₂)