Use fourthorder polynomials P0x and P1x as spline functions

Use fourth–order polynomials P0(x) and P1(x) as spline functions to approximate the function f (x) over the interval of x–values [x0 , x1 , x2 ]. In the interval x0 x x1 , the polynomial P0(x) = a4x^4+a3x^3+a2x^2+a1x+a0 is used while while in the interval x1 x x2, the polynomial P1(x) =b4x^4+b3x^3+b2x^2+b1x+b0 is valid.

a.)Clearly write out the conditions, and resulting equations, that you would use to determine the ten coefficients in P0(x) and P1(x). If needed, you may assume that f(x) and all of its derivatives are known. Explain your choice of conditions used for matching. (You do not need to actually determine the resulting coefficient values a0, a1, . . . , b4.)

Solution

For spline function f(x) over interval [x0,x1,x2]

f\'(x),f\"(x),f\"\'(x) should be continous

so,

at x1

P0(x1)=P1(x1)................(1)

P0\'(x1)=P1\'(x1)................(2)

P0\"(x1)=P\"(x1)............(3)

P0\"\'(x1)=P\"\'(x1)............(4)

by solving above equations and other equations where derivatives are known for x0 and x2 ... one gets ten coefficients of P0(x) and P1(x).