The (n×n) Hilbert matrix is the matrix whose ijth entry is 1/(i +j −1). For example, the (3 × 3) Hilbert matrix is [1 1/2 1/3 1/2 1/3 1/4 1/3 1/4 1/5]. Let A denote the (6 × 6) Hilbert matrix, and consider the vectors b and b + Δb: b = [ 1 2 1 1.414 1 2]. b + Δb = [1 2 1 1.4142 1 2].

Question

The (n×n) Hilbert matrix is the matrix whose ijth entry is 1/(i +j −1). For example, the (3 × 3) Hilbert matrix is

[latex]\left[\begin{array}{ccc}1 & 1 / 2 & 1 / 3 \1 / 2 & 1 / 3 & 1 / 4 \1 / 3 & 1 / 4 & 1 / 5\end{array} ight][/latex].

Let A denote the (6 × 6) Hilbert matrix, and consider the vectors b and b + Δb:

[latex]b =\left[\begin{array}{c}1 \2 \1 \1.414 \1 \2\end{array} ight], \quad b +\Delta b =\left[\begin{array}{c}1 \2 \1 \1.4142 \1 \2\end{array} ight][/latex].

Note that b and b+Δb differ slightly in their fourth components. Compare the solutions of Ax = b and Ax = b + Δb.

Accepted Answer

We used MATLAB to solve these two equations. If [latex]X _{1}[/latex] denotes the solution of Ax = b, and [latex]X _{2}[/latex] denotes the solution of Ax = b + Δb, the results are (rounded to the nearest integer):

[latex]x_{1}=\left[\begin{array}{r}-6538 \185706 \-1256237 \3271363 \-3616326 \1427163\end{array} ight] \quad ext { and } \quad x_{2}=\left[\begin{array}{r}-6539 \185747 \-1256519 \3272089 \-3617120 \1427447\end{array} ight][/latex].

(Note: Despite the fact that b and b + Δb are nearly equal, [latex]x _{1} ext { and } x _{2}[/latex] differ by almost 800 in their fifth components.)

Question 1.9.7: The (n×n) Hilbert matrix is the matrix whose ijth entry is 1...

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