Chapter 1

Linear Equations

[38/84%]
Show
LaTeX
Section 1.2Gaussian Elimination and Matrices[16/94%]
Problem 1.2.1

Use Gaussian elimination with back substitution to solve the following system:

x1+x2+x3=1,x1+2x2+2x3=1,x1+2x2+3x3=1. \begin{aligned} x_{1} + x_{2} + x_{3} & = 1, \\ x_{1} + 2 x_{2} + 2 x_{3} & = 1, \\ x_{1} + 2 x_{2} + 3 x_{3} & = 1. \end{aligned}
?
Problem 1.2.2

Apply Gaussian elimination with back substitution to the following system:

2x1−x2=0,−x1+2x2−x3=0,−x2+x3=1. \begin{aligned} 2 x_{1} - x_{2} & = 0, \\ -x_{1} + 2 x_{2} - x_{3} & = 0, \\ -x_{2} + x_{3} & = 1. \end{aligned}
?
Problem 1.2.3

Use Gaussian elimination with back substitution to solve the following system:

4x2−3x3=3,−x1+7x2−5x3=4,−x1+8x2−6x3=5. \begin{aligned} 4 x_{2} - 3 x_{3} & = 3, \\ -x_{1} + 7 x_{2} - 5 x_{3} & = 4, \\ -x_{1} + 8 x_{2} - 6 x_{3} & = 5. \end{aligned}
?
Problem 1.2.4

Solve the following system:

x1+x2+x3+x4=1,x1+x2+3x3+3x4=3,x1+x2+2x3+3x4=3,x1+3x2+3x3+3x4=4. \begin{aligned} x_{1} + x_{2} + x_{3} + x_{4} & = 1, \\ x_{1} + x_{2} + 3 x_{3} + 3 x_{4} & = 3, \\ x_{1} + x_{2} + 2 x_{3} + 3 x_{4} & = 3, \\ x_{1} + 3 x_{2} + 3 x_{3} + 3 x_{4} & = 4. \end{aligned}
?
Problem 1.2.5

Consider the following three systems where the coefficients are the same for each system, but the right-hand sides are different (this situation occurs frequently):

4x−8y+5z=1∣0∣0,4x−7y+4z=0∣1∣0,3x−4y+2z=0∣0∣1. \begin{aligned} 4x - 8y + 5z & = 1 \mid 0 \mid 0, \\ 4x - 7y + 4z & = 0 \mid 1 \mid 0, \\ 3x - 4y + 2z & = 0 \mid 0 \mid 1. \end{aligned}

Solve all three systems at one time by performing Gaussian elimination on an augmented matrix of the form

[A∣b1∣b2∣b3]. \left[\mathbf{A} \mid \mathbf{b_{1}} \mid \mathbf{b_{2}} \mid \mathbf{b_{3}}\right].
?
Problem 1.2.6

Suppose that matrix BB is obtained by performing a sequence of row operations on matrix AA. Explain why AA can be obtained by performing row operations on BB.

?
Problem 1.2.7

Find angles α\alpha, β\beta, and γ\gamma such that

2sin⁡α−cos⁡β+3tan⁡γ=3,4sin⁡α+2cos⁡β−2tan⁡γ=2,6sin⁡α−3cos⁡β+tan⁡γ=9, \begin{aligned} 2 \sin \alpha - \cos \beta + 3 \tan \gamma & = 3, \\ 4 \sin \alpha + 2 \cos \beta - 2 \tan \gamma & = 2, \\ 6 \sin \alpha - 3 \cos \beta + \tan \gamma & = 9, \end{aligned}

where 0≤α≤2π0 \leq \alpha \leq 2\pi, 0≤β≤2π0 \leq \beta \leq 2\pi, and 0≤γ<π0 \leq \gamma < \pi.

?
Problem 1.2.8

The following system has no solution:

−x1+3x2−2x3=1,−x1+4x2−3x3=0,−x1+5x2−4x3=0. \begin{aligned} -x_{1} + 3 x_{2} - 2 x_{3} & = 1, \\ -x_{1} + 4 x_{2} - 3 x_{3} & = 0, \\ -x_{1} + 5 x_{2} - 4 x_{3} & = 0. \end{aligned}

Attempt to solve this system using Gaussian elimination and explain what occurs to indicate that the system is impossible to solve.

?
Problem 1.2.9

Attempt to solve the system

−x1+3x2−2x3=4,−x1+4x2−3x3=5,−x1+5x2−4x3=6, \begin{aligned} -x_{1} + 3 x_{2} - 2 x_{3} & = 4, \\ -x_{1} + 4 x_{2} - 3 x_{3} & = 5, \\ -x_{1} + 5 x_{2} - 4 x_{3} & = 6, \end{aligned}

using Gaussian elimination and explain why this system must have infinitely many solutions.

?
Problem 1.2.10

By solving a 3×33 \times 3 system, find the coefficients in the equation of the parabola y=α+βx+γx2y = \alpha + \beta x + \gamma x^{2} that passes through the points (1,1)(1, 1), (2,2)(2, 2), and (3,0)(3, 0).

?
Problem 1.2.11

Suppose that 100 insects are distributed in an enclosure consisting of four chambers with passageways between them as shown below.

At the end of one minute, the insects have redistributed themselves. Assume that a minute is not enough time for an insect to visit more than one chamber and that at the end of a minute 40 of the insects in each chamber have not left the chamber they occupied at the beginning of the minute. The insects that leave a chamber disperse uniformly among the chambers that are directly accessible from the one they initially occupied—e.g., from 3, half move to 2 and half move to 4.

?
(a)

If at the end of one minute there are 12, 25, 26, and 37 insects in chambers 1, 2, 3, and 4, respectively, determine what the initial distribution had to be.

(b)

If the initial distribution is 20, 20, 20, 40, what is the distribution at the end of one minute?

Problem 1.2.12

Show that the three types of elementary row operations discussed on p.8 are not independent by showing that the interchange operation (1.2.7) can be accomplished by a sequence of the other two types of row operations given in (1.2.8) and (1.2.9).

Recall the three elementary row operations: Type I (equation (1.2.7)), interchange the order of two equations; Type II, multiply an equation by a nonzero scalar; Type III, replace an equation by the sum of itself and a multiple of another equation. Equations (1.2.8) and (1.2.9) are the Type II and Type III operations respectively.

?
Problem 1.2.13

Suppose that [A∣b][A \mid b] is the augmented matrix associated with a linear system. You know that performing row operations on [A∣b][A \mid b] does not change the solution of the system. However, no mention of column operations was ever made because column operations can alter the solution.

?
(a)

Describe the effect on the solution of a linear system when columns A∗jA_{*j} and A∗kA_{*k} are interchanged.

(b)

Describe the effect when column A∗jA_{*j} is replaced by αA∗j\alpha A_{*j} for α≠0\alpha \neq 0.

(c)

Describe the effect when A∗jA_{*j} is replaced by A∗j+αA∗kA_{*j} + \alpha A_{*k}.

Remark.
?

Experiment with a 2×22 \times 2 or 3×33 \times 3 system.

Problem 1.2.14

Consider the n×nn \times n Hilbert matrix defined by

H=[11213⋯1n]2mm]121314⋯1n+1131415⋯1n+2]2mm]⋮⋮⋮⋯⋮1n1n+11n+2⋯12n−1]. H = \begin{bmatrix} 1 & \dfrac {1}{2} & \dfrac {1}{3} & \cdots & \dfrac {1}{n} \\ ]2mm] \dfrac {1}{2} & \dfrac {1}{3} & \dfrac {1}{4} & \cdots & \dfrac {1}{n+1} \\[2mm] \dfrac {1}{3} & \dfrac {1}{4} & \dfrac {1}{5} & \cdots & \dfrac {1}{n+2} \\ ]2mm] \vdots & \vdots & \vdots & \cdots & \vdots \\[2mm] \dfrac {1}{n} & \dfrac {1}{n+1} & \dfrac {1}{n+2} & \cdots & \dfrac {1}{2n-1} \end{bmatrix}.

Express the individual entries hijh_{ij} in terms of ii and jj.

?
Problem 1.2.15

Verify that the operation counts given in the text for Gaussian elimination with back substitution are correct for a general 3×33 \times 3 system. If you are up to the challenge, try to verify these counts for a general n×nn \times n system.

The operation counts given in the text: Gaussian elimination with back substitution applied to an n×nn \times n system requires n33+n2−n3\dfrac {n^{3}}{3} + n^{2} - \dfrac {n}{3} multiplications/divisions and n33+n22−5n6\dfrac {n^{3}}{3} + \dfrac {n^{2}}{2} - \dfrac {5n}{6} additions/subtractions.

?
Problem 1.2.16

Explain why a linear system can never have exactly two different solutions. Extend your argument to explain the fact that if a system has more than one solution, then it must have infinitely many different solutions.

?
Section 1.3Gauss--Jordan Method[4/75%]
Problem 1.3.1

Use the Gauss--Jordan method to solve the following system:

4x2−3x3=3,−x1+7x2−5x3=4,−x1+8x2−6x3=5. \begin{aligned} 4 x_{2} - 3 x_{3} & = 3, \\ -x_{1} + 7 x_{2} - 5 x_{3} & = 4, \\ -x_{1} + 8 x_{2} - 6 x_{3} & = 5. \end{aligned}
?
Problem 1.3.2

Apply the Gauss--Jordan method to the following system:

x1+x2+x3+x4=1,x1+2x2+2x3+2x4=0,x1+2x2+3x3+3x4=0,x1+2x2+3x3+4x4=0. \begin{aligned} x_{1} + x_{2} + x_{3} + x_{4} & = 1, \\ x_{1} + 2 x_{2} + 2 x_{3} + 2 x_{4} & = 0, \\ x_{1} + 2 x_{2} + 3 x_{3} + 3 x_{4} & = 0, \\ x_{1} + 2 x_{2} + 3 x_{3} + 4 x_{4} & = 0. \end{aligned}
?
Problem 1.3.3

Use the Gauss--Jordan method to solve the following three systems at the same time.

2x1−x2=1∣0∣0,−x1+2x2−x3=0∣1∣0,−x2+x3=0∣0∣1. \begin{aligned} 2 x_{1} - x_{2} & = 1 \mid 0 \mid 0, \\ -x_{1} + 2 x_{2} - x_{3} & = 0 \mid 1 \mid 0, \\ -x_{2} + x_{3} & = 0 \mid 0 \mid 1. \end{aligned}
?
Problem 1.3.4

Verify that the operation counts given in the text for the Gauss--Jordan method are correct for a general 3×33 \times 3 system. If you are up to the challenge, try to verify these counts for a general n×nn \times n system.

The operation counts given in the text: for an n×nn \times n system, the Gauss--Jordan procedure requires n32+n22\dfrac {n^{3}}{2} + \dfrac {n^{2}}{2} multiplications/divisions and n32−n2\dfrac {n^{3}}{2} - \dfrac {n}{2} additions/subtractions.

?
Section 1.4Two-Point Boundary Value Problems[3/33%]
Problem 1.4.1

Divide the interval [0,1][0, 1] into five equal subintervals, and apply the finite difference method in order to approximate the solution of the two-point boundary value problem

y′′(t)=125t,y(0)=y(1)=0 y''(t) = 125 t, \quad y(0) = y(1) = 0

at the four interior grid points. Compare your approximate values at the grid points with the exact solution at the grid points. Note: You should not expect very accurate approximations with only four interior grid points.

?
Problem 1.4.2

Divide [0,1][0, 1] into n+1n+1 equal subintervals, and apply the finite difference approximation method to derive the linear system associated with the two-point boundary value problem

y′′(t)−y′(t)=f(t),y(0)=y(1)=0. y''(t) - y'(t) = f(t), \quad y(0) = y(1) = 0.
?
Problem 1.4.3

Divide [0,1][0, 1] into five equal subintervals, and approximate the solution to

y′′(t)−y′(t)=125t,y(0)=y(1)=0 y''(t) - y'(t) = 125 t, \quad y(0) = y(1) = 0

at the four interior grid points. Compare the approximations with the exact values at the grid points.

?
Section 1.5Making Gaussian Elimination Work[8/100%]
Problem 1.5.1

Consider the following system:

10−3x−y=1,x+y=0. \begin{aligned} 10^{-3} x - y & = 1, \\ x + y & = 0. \end{aligned}
?
(a)

Use 3-digit arithmetic with no pivoting to solve this system.

(b)

Find a system that is exactly satisfied by your solution from part (a), and note how close this system is to the original system.

(c)

Now use partial pivoting and 3-digit arithmetic to solve the original system.

(d)

Find a system that is exactly satisfied by your solution from part (c), and note how close this system is to the original system.

(e)

Use exact arithmetic to obtain the solution to the original system, and compare the exact solution with the results of parts (a) and (c).

(f)

Round the exact solution to three significant digits, and compare the result with those of parts (a) and (c).

Problem 1.5.2

Consider the following system:

x+y=3,−10x+105y=105. \begin{aligned} x + y & = 3, \\ -10 x + 10^{5} y & = 10^{5}. \end{aligned}
?
(a)

Use 4-digit arithmetic with partial pivoting and no scaling to compute a solution.

(b)

Use 4-digit arithmetic with complete pivoting and no scaling to compute a solution of the original system.

(c)

This time, row scale the original system first, and then apply partial pivoting with 4-digit arithmetic to compute a solution.

(d)

Now determine the exact solution, and compare it with the results of parts (a), (b), and (c).

Problem 1.5.3

With no scaling, compute the 3-digit solution of

−3x+y=−2,10x−3y=7, \begin{aligned} -3 x + y & = -2, \\ 10 x - 3 y & = 7, \end{aligned}

without partial pivoting and with partial pivoting. Compare your results with the exact solution.

?
Problem 1.5.4

Consider the following system in which the coefficient matrix is the Hilbert matrix:

x+12y+13z=13,12x+13y+14z=13,13x+14y+15z=15. \begin{aligned} x + \frac{1}{2} y + \frac{1}{3} z & = \frac{1}{3}, \\ \frac{1}{2} x + \frac{1}{3} y + \frac{1}{4} z & = \frac{1}{3}, \\ \frac{1}{3} x + \frac{1}{4} y + \frac{1}{5} z & = \frac{1}{5}. \end{aligned}
?
(a)

First convert the coefficients to 3-digit floating-point numbers, and then use 3-digit arithmetic with partial pivoting but with no scaling to compute the solution.

(b)

Again use 3-digit arithmetic, but row scale the coefficients (after converting them to floating-point numbers), and then use partial pivoting to compute the solution.

(c)

Proceed as in part (b), but this time row scale the coefficients before each elimination step.

(d)

Now use exact arithmetic on the original system to determine the exact solution, and compare the result with those of parts (a), (b), and (c).

Problem 1.5.5

To see that changing units can affect a floating-point solution, consider a mining operation that extracts silica, iron, and gold from the earth. Capital (measured in dollars), operating time (in hours), and labor (in man-hours) are needed to operate the mine. To extract a pound of silica requires $.0055, .0011 hours of operating time, and .0093 man-hours of labor. For each pound of iron extracted, $.095, .01 operating hours, and .025 man-hours are required. For each pound of gold extracted, $960, 112 operating hours, and 560 man-hours are required.

?
(a)

Suppose that during 600 hours of operation, exactly $5000 and 3000 man-hours are used. Let xx, yy, and zz denote the number of pounds of silica, iron, and gold, respectively, that are recovered during this period. Set up the linear system whose solution will yield the values for xx, yy, and zz.

(b)

With no scaling, use 3-digit arithmetic and partial pivoting to compute a solution (x~,y~,z~)(\tilde{x}, \tilde{y}, \tilde{z}) of the system of part (a). Then approximate the exact solution (x,y,z)(x, y, z) by using your machine's (or calculator's) full precision with partial pivoting to solve the system in part (a), and compare this with your 3-digit solution by computing the relative error defined by

er=(x−x~)2+(y−y~)2+(z−z~)2x2+y2+z2. e_{r} = \frac{\sqrt{(x-\tilde{x})^{2}+(y-\tilde{y})^{2}+(z-\tilde{z})^{2}}}{\sqrt{x^{2}+y^{2}+z^{2}}}.
(c)

Using 3-digit arithmetic, column scale the coefficients by changing units: convert pounds of silica to tons of silica, pounds of iron to half-tons of iron, and pounds of gold to troy ounces of gold (1 lb. = 12 troy oz.).

(d)

Use 3-digit arithmetic with partial pivoting to solve the column scaled system of part (c). Then approximate the exact solution by using your machine's (or calculator's) full precision with partial pivoting to solve the system in part (c), and compare this with your 3-digit solution by computing the relative error ere_{r} as defined in part (b).

Problem 1.5.6

Consider the system given in Example 1.5.3.

Example 1.5.3's system: x−y=−2x-y=-2, −9x+10y=12-9x+10y=12.

?
(a)

Use 3-digit arithmetic with partial pivoting but with no scaling to solve the system.

(b)

Now use partial pivoting with scaling. Does complete pivoting provide an advantage over scaled partial pivoting in this case?

Problem 1.5.7

Consider the following well-scaled matrix:

Wn=[100⋯001−110⋯001−1−11⋱001⋮⋮⋱⋱⋱⋮⋮−1−1−1⋱101−1−1−1⋯−111−1−1−1⋯−1−11]. W_{n} = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 & 0 & 1 \\ -1 & 1 & 0 & \cdots & 0 & 0 & 1 \\ -1 & -1 & 1 & \ddots & 0 & 0 & 1 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ -1 & -1 & -1 & \ddots & 1 & 0 & 1 \\ -1 & -1 & -1 & \cdots & -1 & 1 & 1 \\ -1 & -1 & -1 & \cdots & -1 & -1 & 1 \end{bmatrix}.
?
(a)

Reduce WnW_{n} to an upper-triangular form using Gaussian elimination with partial pivoting, and determine the element of maximal magnitude that emerges during the elimination procedure.

(b)

Now use complete pivoting and repeat part (a).

(c)

Formulate a statement comparing the results of partial pivoting with those of complete pivoting for WnW_{n}, and describe the effect this would have in determining the tt-digit solution for a system whose augmented matrix is [Wn∣b][W_{n} \mid b].

Problem 1.5.8

Suppose that AA is an n×nn \times n matrix of real numbers that has been scaled so that each entry satisfies ∣aij∣≤1\left|a_{ij}\right| \leq 1, and consider reducing AA to triangular form using Gaussian elimination with partial pivoting. Demonstrate that after kk steps of the process, no entry can have a magnitude that exceeds 2k2^{k}.

?
Remark.
?

The previous exercise shows that there are cases where it is possible for some elements to actually attain the maximum magnitude of 2k2^{k} after kk steps.

Section 1.6Ill-Conditioned Systems[7/71%]
Problem 1.6.1

Consider the ill-conditioned system of Example 1.6.1:

.835x+.667y=.168,.333x+.266y=.067. \begin{aligned} .835 x + .667 y & = .168, \\ .333 x + .266 y & = .067. \end{aligned}
?
(a)

Describe the outcome when you attempt to solve the system using 5-digit arithmetic with no scaling.

(b)

Again using 5-digit arithmetic, first row scale the system before attempting to solve it. Describe to what extent this helps.

(c)

Now use 6-digit arithmetic with no scaling. Compare the results with the exact solution.

(d)

Using 6-digit arithmetic, compute the residuals for your solution of part (c), and interpret the results.

(e)

For the same solution obtained in part (c), again compute the residuals, but use 7-digit arithmetic this time, and interpret the results.

(f)

Formulate a concluding statement that summarizes the points made in parts (a)--(e).

Problem 1.6.2

Perturb the ill-conditioned system given in Exercise 1.6.1 above so as to form the following system:

.835x+.667y=.1669995,.333x+.266y=.066601. \begin{aligned} .835 x + .667 y & = .1669995, \\ .333 x + .266 y & = .066601. \end{aligned}
?
(a)

Determine the exact solution, and compare it with the exact solution of the system in Exercise 1.6.1.

(b)

On the basis of the results of part (a), formulate a statement concerning the necessity for the solution of an ill-conditioned system to undergo a radical change for every perturbation of the original system.

Problem 1.6.3

Consider the two straight lines determined by the graphs of the following two equations:

.835x+.667y=.168,.333x+.266y=.067. \begin{aligned} .835 x + .667 y & = .168, \\ .333 x + .266 y & = .067. \end{aligned}
?
(a)

Use 5-digit arithmetic to compute the slopes of each of the lines, and then use 6-digit arithmetic to do the same. In each case, sketch the graphs on a coordinate system.

(b)

Show by diagram why a small perturbation in either of these lines can result in a large change in the solution.

(c)

Describe in geometrical terms the situation that must exist in order for a system to be optimally well-conditioned.

Problem 1.6.4

Using geometric considerations, rank the following three systems according to their condition.

?
(a)
1.001x−y=.235,x+.0001y=.765. \begin{aligned} 1.001 x - y & = .235, \\ x + .0001 y & = .765. \end{aligned}
(b)
1.001x−y=.235,x+.9999y=.765. \begin{aligned} 1.001 x - y & = .235, \\ x + .9999 y & = .765. \end{aligned}
(c)
1.001x+y=.235,x+.9999y=.765. \begin{aligned} 1.001 x + y & = .235, \\ x + .9999 y & = .765. \end{aligned}
Problem 1.6.5

Determine the exact solution of the following system:

8x+5y+2z=15,21x+19y+16z=56,39x+48y+53z=140. \begin{aligned} 8 x + 5 y + 2 z & = 15, \\ 21 x + 19 y + 16 z & = 56, \\ 39 x + 48 y + 53 z & = 140. \end{aligned}

Now change 1515 to 1414 in the first equation and again solve the system with exact arithmetic. Is the system ill-conditioned?

?
Problem 1.6.6

Show that the system

v−w−x−y−z=0,w−x−y−z=0,x−y−z=0,y−z=0,z=1, \begin{aligned} v - w - x - y - z & = 0, \\ w - x - y - z & = 0, \\ x - y - z & = 0, \\ y - z & = 0, \\ z & = 1, \end{aligned}

is ill-conditioned by considering the following perturbed system:

v−w−x−y−z=0,−115v+w−x−y−z=0,−115v+x−y−z=0,−115v+y−z=0,−115v+z=1. \begin{aligned} v - w - x - y - z & = 0, \\ -\frac{1}{15} v + w - x - y - z & = 0, \\ -\frac{1}{15} v + x - y - z & = 0, \\ -\frac{1}{15} v + y - z & = 0, \\ -\frac{1}{15} v + z & = 1. \end{aligned}
?
Problem 1.6.7

Let f(x)=sin⁡πxf(x) = \sin \pi x on [0,1][0, 1]. The object of this problem is to determine the coefficients αi\alpha_{i} of the cubic polynomial

p(x)=∑i=03αixi p(x) = \sum _{i=0}^{3} \alpha _{i} x^{i}

that is as close to f(x)f(x) as possible in the sense that

r=∫01[f(x)−p(x)]2 dx=∫01[f(x)]2 dx−2∑i=03αi∫01xif(x) dx+∫01(∑i=03αixi)2dx \begin{aligned} r & = \int _{0}^{1} [f(x) - p(x)]^{2} \, dx \\ & = \int _{0}^{1} [f(x)]^{2} \, dx - 2 \sum _{i=0}^{3} \alpha _{i} \int _{0}^{1} x^{i} f(x) \, dx + \int _{0}^{1} \left( \sum _{i=0}^{3} \alpha _{i} x^{i} \right)^{2} dx \end{aligned}

is as small as possible.

?
(a)

In order to minimize rr, impose the condition that ∂r/∂αi=0\partial r / \partial \alpha_{i} = 0 for each i=0,1,2,3i = 0, 1, 2, 3, and show this results in a system of linear equations whose augmented matrix is [H4∣b][H_{4} \mid b], where H4H_{4} and bb are given by

H4=[1121314]2mm]1213141513141516]2mm]14151617]andb=[2π1π]2mm]1π−4π31π−6π3]. H_{4} = \begin{bmatrix} 1 & \dfrac {1}{2} & \dfrac {1}{3} & \dfrac {1}{4} \\ ]2mm] \dfrac {1}{2} & \dfrac {1}{3} & \dfrac {1}{4} & \dfrac {1}{5} \\[2mm] \dfrac {1}{3} & \dfrac {1}{4} & \dfrac {1}{5} & \dfrac {1}{6} \\ ]2mm] \dfrac {1}{4} & \dfrac {1}{5} & \dfrac {1}{6} & \dfrac {1}{7} \end{bmatrix} \quad \text{and} \quad b = \begin{bmatrix} \dfrac {2}{\pi } \\[2mm] \dfrac {1}{\pi } \\ ]2mm] \dfrac {1}{\pi } - \dfrac {4}{\pi ^{3}} \\[2mm] \dfrac {1}{\pi } - \dfrac {6}{\pi ^{3}} \end{bmatrix}.

Any matrix HnH_{n} that has the same form as H4H_{4} is called a Hilbert matrix of order nn.

(b)

Systems involving Hilbert matrices are badly ill-conditioned, and the ill-conditioning becomes worse as the size increases. Use exact arithmetic with Gaussian elimination to reduce H4H_{4} to triangular form. Assuming that the case in which n=4n = 4 is typical, explain why a general system [Hn∣b][H_{n} \mid b] will be ill-conditioned. Notice that even complete pivoting is of no help.