MTC7 - Autumn 2019
Final Exam - January, the 17th 2020
Exercise 1 LetI =]a, b[ andJ two intervals of Rand f :J×I →R. Give the conditions on the functionf such that the integral depending on a parameter
F(x) = Z b
a
f(x, t) dt
converges and that the functionF is continuous overJ.
Exercise 2 Prove the statement : For f a continuous, non-negative and decreasing function on the interval [1,+∞[, let us pose uk =f(k) for all integer k≥1,then
Xuk converges ⇐⇒
Z ∞
1
f(x) dx converges.
Exercise 3In some cases, an invertible n×nmatrixA∈Rn×n and a vectorb∈Rn are such that the solutionx∈Rn of the linear system
Ax=b
is not stable under small perturbations of b. For instance forA=
−32 −66 66 133
and b= 40
−80
the solution is x =
0.4
−0.8
. If we perturb slightly the right-hand side, b0 =
40.2
−80.1
the new solution turns out to bex0 =
0.6
−0.9
. In other words, a relative error of the order 0.2/40 = 5×10−3 and 0.1/80 = 1.25×10−3 in the databproduces a relative error of the orders (0.6−0.4)/0.4 = 5×10−1 and in the first component of the solution, which represents an amplification of the relative error of the order 5×10−1/(5×10−3) = 100. In this exercise you will discover the reason of this phenomena, and in the same time the spectral norm||.||∞ of matrices.
a. Posingb0 =b+δb and x0 =x+δx prove that Aδx=δb.
b. Let ||x||=p
x21+x22 be the euclidean norm of any vector x= (x1, x2)T ∈R2 and let the circle C={x∈R2 | ||x||= 1}. Prove that if||Ax||= 0 for allx∈C thenA is the null matrix.
c. Let M2(R) be the vector space of the 2×2 matrices and the mapping
||.||∞: M2(R) → R
A 7→ ||A||∞= supx∈C||Ax|| .
Prove that||.||∞defines a norm over M2(R). To prove the triangle inequality we admit the property of the suppremum over a setE: supx∈E(g(x) +h(x))≤supx∈Eg(x) + supx∈Eh(x) for any functions gand h:E →R.
d.Prove that the sum of the dimensions of the eigenspaces of a diagonalizable and invertible matrix A∈ M2(R) is equal to two and that its eigenvalues are non vanishing.
In the following, we consider only diagonalizable and invertible matrices A, we denoteλ1 and λ2 the eigenvalues corresponding to their two independent eigenvectorsU1 andU2 and we choose 0<|λ1| ≤
|λ2|.
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e. Prove that ||A||∞ = |λ2|. We suggest to use, after its justification, the decomposition x = aU1+bU2.
f. Express the eigenvalues of A−1 in terms of λ1 andλ2 to prove that||A−1||∞= 1
|λ1|.
g. Establish that||By|| ≤ ||B||∞ ||y||for any y∈R2 and any diagonalizable and invertible matrix B to prove that
||δx||
||x|| ≤ ||A||∞||A−1||∞
||δb||
||b|| . An intermediary step is the proof of ||x||1 ≤ ||A−1||∞||b||1 .
h. In the introductory example the matrix A is invertible and diagonalizable with λ1 = 1 and λ2 = 100. Explain the instability result using the ratio ||A||∞||A−1||∞ = |λ2|
|λ1| called the condition number of the matrix A and propose a condition that garanties a small instability in the resolution of a 2×2 linear system.
Exercise 4Let a sequence of functions un:x7→ (−1)n+1
(n+x)2 forn∈N and x∈]0,+∞[.
a. Show that the series
∞
P
n=0
un(x) simply converges on ]0,+∞[.
b. Prove that the series of the derivatives
∞
P
n=0
u0n(x) is normally convergent on ]0,+∞[.
c. Find the expression of the derivative S0(x) of the sum S(x) =
∞
P
n=0
un(x) under the form of a series. A detailed justification is required.
Exercise 5A door is opened between two rooms that holdv(0) = 30 people andw(0) = 10 people.
The movement between rooms is assumed to be proportional to the differencev−w:
dv
dt =w−v and dw
dt =v−w.
a. Without computing the solution, show that the total numberv+wof people in the two rooms is constant (40 people).
b. Write the system under the form of a system of differential equation du
dt =Au with matrix A, solve this system and findv and w att= 1.
Exercise 6 Show but without computation that the minimization problem
a, b∈minR
Z 1
0
(e2t−aet−b)2dt
can be expressed as a distance. Be very precise on the definition of this distance.
Exercise 7 Justify in detail the existence of the limit
n→∞lim Z ∞
0
tn 1 +tn+2dt and calculate its value.
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