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

2
3

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

>৉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
ৗ৉
ে)৘৉
*

i) 假设上述求积法则对所有 ӑ3 次多项式都是准确的，

, ৉ > 2- 3- 4- 并证明假设

[8 分]
ii) 给定第 i) 部分中的权重公式，令 ৘2 >ౖ, ৘3 > 2
3

)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]