Numerical methods numerical analysis Questions and Answers

Numerical methods numerical analysis Questions and Answers Theory of regression analysis - Answer-A statistical technique for estimating relationship among variables Piecewise linear interpolation - Answer-Is the simplest form of curve fitting - need to find equations to represent the piecewise linear functions. Do this by finding linear function between the two data points and then also need to find linear function between each two adjacent data points Limitations of piecewise linear interpolation (4) - Answer-1) simple model - 1st order polynomial 2) practically less useable since most data can not be represented using linear functions 3) sharp change in the first derivative at data points 4) higher order polynomials are often used in practical problems so can't use this for them Polynomial fits - Answer-Polynomials used to model data more accurately Theory of polynomial fit - Answer-Use a matrix to represent data set Advantages of using polynomial fit to represent and analyse data (4) - Answer-1) simple model 2) polynomials are smooth functions 3) polynomial of degree n-1 can be represented exactly with a set of n coefficients 4) if f(x) is any continuous function defined over any finite interval [a,b] then for each e, ther the exists a polynomial fit such that |f(x) - F(x)| e for all a x b Limitations of polynomial fit - Answer-When higher order polynomials are used the output can be very sensitive to the changes in a single or a few terms only - eg. x^5 is too significant and will decide the value. May then oscillate widely between adjacent data points - known as ill condition f polynomials Lagrange polynomials - Answer-Instead of using general form of polynomials can use alternative polynomial system Lagrange polynomials - is a way to avoid effect of ill conditioned polynomials