numerical methods 3

Legendre Polynomials

Legendre polynomials are a family of degree-n polynomials that are orthogonal on the interval (−1,1)(-1,1)(−1,1).

Monic

leading coefficient is one

orthogonality property

P(x) is any polynomial of degree less than n. p(n) is the degree n Legendre polynomial. integral from -1 to 1 of their product is 0.

ODE-IVP

ordinary differential equation initial value problems

well-posed

a problem where a solution exists, is unique, and depends continuously on the problem data (small changes in data → small changes in the solution)

lipschitz condition

|f(t, y1)-f(t, y2) <= L|y1-y2|

lipschitz constant

The positive constant L that makes the Lipschitz condition true.

one-step method

approximation at the current step depends only on the approximation at the one previous step

multistep method

calculations for the next value require the m previous values for an m-step.

explicit

reliant on only one variable (can isolate y)

implicit

reliant on x and y.

predictor-corrector method

explicit methods used in pairs with implicit methods of the same order for higher accuracy.

consistent

the local truncation error converges to zero as the step size goes to zero.

stable

small changes in the problem data cause corresponding small changes in approximations at each time step.

convergent

the global truncation error at each time step converges to zero as the step size goes to zero.

quadrature

estimating integrals.