本次英国作业主要为计算相关的数学代写限时测试
1. (a) i) 考虑点 >υ 在网格上的完整三次样条插值，
> ѿ2- 1- 2- 3- 4, υ> 2
3
, 插入一个足够平滑的函数 ে 在
点 >1,
2
3
和 2. 导出 B 样条的线性方程组
系数ৄ, > ѿ2- 1- 2- 3- 4- 定义样条插值。它不是
需要解决这个系统。
提示：标准 প3
3 次 B 样条由下式给出：
ii) 说明完整三次样条插值的误差的收敛结果
使用网格在区间 \1- 2^ 上插入一个足够平滑的函数 ে
与 等距网格点
>υ, > 1- 2- 3-Ϳ-- 其中 υ > 2。
[4 分]
(b) 考虑近似
2
1
ে)*e 使用三点求积法则
4
>2
ৗ
ে)
*
哪里 ৗ
和
是定义正交规则的权重和点。
i) 假设上述求积法则对所有 ӑ3 次多项式都是准确的，
陈述一个定义权重的公式
, > 2- 3- 4- 并证明假设
确实对那个公式很满意。
提示：考虑拉格朗日多项式。
[8 分]
ii) 给定第 i) 部分中的权重公式，令 2 >ౖ, 3 > 2
3
和 4 >
)2 ѿ ౖ*, 对于某些 ౖӍ 2
3
.证明如此定义的求积法则对所有
4 次多项式。
[5 分]
1. (a) i) Consider the complete cubic spline interpolant on the grid with points >υ,
> ѿ2- 1- 2- 3- 4, υ> 2
3
, that interpolates a sufficiently smooth function ে at the
points >1,
2
3
and 2. Derive the system of linear equations for the B-spline
coefficients ৄ, > ѿ2- 1- 2- 3- 4- which define the spline interpolant. It is not
required to solve this system.
Hint: The standard প3
B-spline of degree 3 is given by:
ii) State a convergence result for the error of the complete cubic spline interpolant
that interpolates a sufficiently smooth function ে on the interval \1- 2^ using a mesh
with equally-spaced grid-points
>υ, > 1- 2- 3-Ϳ-- where υ > 2.
[4 marks]
(b) Consider the approximation of
2
1
ে)*e using the three-point quadrature rule
4
า >2
ৗ
ে)
*
where ৗ
and
are the weights and points defining the quadrature rule.
i) Assuming the above quadrature rule is exact for all polynomials of degree ӑ3,
state a formula defining the weights ৗ
, > 2- 3- 4- and show that the assumption
is indeed satisfied for that formula.
Hint: Consider Lagrange polynomials.
[8 marks]
ii) Given the formula for the weights found in part i), let 2 >ౖ, 3 > 2
3
and 4 >
)2 ѿ ౖ*, for some ౖӍ 2
3
. Show that the so-defined quadrature rule is exact for all
polynomials of degree ӑ4.
[5 marks]