First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 5. APPROXIMATION THEORY 198
Name Number of Occurrences
Smith 2,376,206
Johnson 1,857,160
Williams 1,534,042
Brown 1,380,145
Jones 1,362,755
Miller 1,127,803
Davis 1,072,335
Garcia 858,289
Rodriguez 804,240
Wilson 783,051
5.2 Continuous least squares
In discrete least squares, our starting point was a set of data points. Here we will start with
a continuous function f on [a, b] and answer the following question: how can we find the
"best" polynomial P n (x) = ∑ n
j=0 a jx j of degree at most n, that approximates f on [a, b]? As
before, "best" polynomial will mean the polynomial that minimizes the least squares error:
E =
∫ b
a
(
f(x) −
Compare this expression with that of the discrete least squares:
E =
To minimize E in (5.10) weset ∂E
∂a k
∂E
∂a k
=
⎛
∂ ⎝
∂a k
= −2
∫ b
a
∫ b
(
m∑
y i −
i=1
) 2 n∑
a j x j dx. (5.10)
j=0
) 2 n∑
a j x j i .
j=0
=0, for k =0, 1, ..., n, and observe
∫ (
b
n∑
)
f 2 (x)dx − 2 f(x) a j x j dx +
a
a
j=0
n∑
∫ b
f(x)x k dx +2 a j x j+k dx =0,
j=0
a
∫ b
a
( n∑
j=0
⎞
) 2
a j x j dx⎠