please solve step by step MAT 362 Numerical Analysis Define

please solve step by step... MAT 362 Numerical Analysis

Define a cubic spline fitting the data {(xi, f(x))}i=0,.. n. particular, use notation to define the 4n unknowns, and supply the 4n equations needed to solve for the unknowns. How is a tridiagonal solver used in solving for these unknowns?

Solution

We consider the case X = [a, b]. In spline interpolation, the
interval [a, b] is partitioned into n smaller subintervals [xi?1, xi
]
by n + 1 interpolation nodes xi
, i = 0 : n. Here we let the
index start with 0, for convenience. A spline s(x) of degree d is
a piece-wise polynomial in C
d?1
, namely,
1. piecewise polynomial
On each [xi?1, xi
], S(x) is a polynomial of degree ? d,
s(x) = pi?1(x), x ? [xi?1, xi
], i = 0 : n ? 1.

We consider the case X = [a, b]. In spline interpolation, the
interval [a, b] is partitioned into n smaller subintervals [xi?1, xi
]
by n + 1 interpolation nodes xi
, i = 0 : n. Here we let the
index start with 0, for convenience. A spline s(x) of degree d is
a piece-wise polynomial in C
d?1
, namely,
1. piecewise polynomial
On each [xi?1, xi
], S(x) is a polynomial of degree ? d,
s(x) = pi?1(x), x ? [xi?1, xi
], i = 0 : n ? 1.