hilbert矩阵

O We first calculate some of its condition number
N Cond (N)
1
2 3 4 5
6
O The relation of N and cond(N) is shown as:
//画一张N和cond（N）的关系表（就用上一张表 的数据（N取1~6））
O Suppose a continuous function f(x) is given on
the interval [0,1] and we are asked to approximate f(x) by a polynomial of degree n - 1 in x. We write the polynomial in the form and define the error in the approximation to be
梁延潇 侯思宇 阮郑阳
Introduction Analysis
Assumption
Test Conclusion
O What is Hilbert Matrix
A matrix with element of line i, row j is 1/(i+j-1) .
Hn
1 1 2 1 n
O -use [不知道matlab里是什么函数] directly to
calculate cond[N] //贴图：数据和图表，忽悠N张幻灯片
O
O The result coincide with our assumption
accurately so we reach the conclusion The Matrix is badly conditioned, Cond(N) grows exponentially when N increases with the formula: cond(N)=e^();
O Thus the column of coefficients C={C1,C2, …Cn}T
can be found by solving the system n×n system where the matrix has elements and the vector b={b1,b2…bn}T is determined by the given function f(x) The matrix Hn is the n×n Hilbert matrix.
O The coห้องสมุดไป่ตู้fficients Ci are determined by the
requirement that E be minimized. Since the error is a differentiable function of the unknowns Ci , at the minimum O Evaluating these derivatives leads to the conditions
O Interchanging the summation and integration, we
obtain
O There are n equations to be satisfied by the
n unknowns Ci . If we let
then the equations can be written as:
1 2 1 3
1 n 1
1 2 n 1
1
n
摘自姥姥课件
O How it comes into being ?
Hilbert (1894) introduced the Hilbert matrix to study the following question in approximation theory: "Assume that I = [a, b] is a real interval. Is it then possible to find a non-zero polynomial P with integral coefficients, such that the integral is smaller than any given bound ε > 0, taken arbitrarily small?"
O As the data shown above, we assume a
model for the relation of N and the cond(N): O Cond(N)=e^(a*N+b) O We use two ways to calculate the condition number in matlab to see how it grows while N increases.