math 502a
homework代写 | express作业 | 代做math | mining – 本题是一个利用math进行练习的代做, 对math的流程进行训练解析, 是比较有代表性的math等代写方向, 这是值得参考的homework代写的题目
math 502a, Spring 2022
homework 6
Due Date:
Problem 1.
(a) Consider the function
f(t) = 0.05 sin(1000t) + 0.5 cos(t) 0 .4 sin(10t)
(i) Evaluatefat the pointsti= 0 + 0. 01 i,i= 0, 1 ,...,100 and plot the resulting
piecewise linear approximation to the function.
(ii) In order to study the slow scale trend of this function, we wish to find a low
degree polynomial (degree at most 6) that best approximatesf in the least
squares sense. What would be the smallest value ofnthat we could expect to
produce a good approximation?
(iii) Using the 101 data points, find the best approximating polynomial of degreen
and plot it along withf.
(b) Let{k}nk=1benlinearly independent real-valued functions inL^2 (a,b) and letQbe
thennmatrix with entries
Qij=
b
a
i(x)j(x)dx.
(i) Show thatQis positive definite and symmetric and therefore invertible.
(ii) Letgbe a real-valued function inL^2 (a,b). What is the best approximation tog
in Span{k}nk=1?
(iii) Consider the case when (a,b) = (0,),n= 4,
k(x) =xk^1 and g(x) = sinx
Find the best approximation ofgin Span{k}^4 k=1and plot the graphs ofgand
its best approximation.
Problem 2.Consider the constrained least squares problem:
min
{x:Cx=d}
||Axb|| 2
where themnmatrixA, thepnmatrixC, and the vectorsbCmanddCpare given.
(a) Show that the unconstrained least squares problem
min
xCn
||Axb|| 2
is a special case of the constrained least squares problem above.
1
2
(b) Show that the minimum norm problem
min
{x:Cx=d}
||x|| 2
is a special case of the constrained least squares problem above.
(c) By writing x= x 0 +Nz show that solving the constrained least squares problem
above is equivalent to solving an unconstrained least squares problem:
min
zCk
||Az b|| 2.
What are the matricesN andA , the vectorsx 0 and b, and the integerk?
(d) Use part (c) to solve the constrained least squares problem where
A=
0 1 2
1 4 3
0 0 2
1 2 4
0 0 1
, b=
3
3
2
2
1
, C=
4 16 12
2 8 6
1 4 3
1 4 3
, d=
12
6
3
3
Problem 3.SupposeAis anmnmatrix withm > nandbRm. Suppose rank(A)< n so a solution to the least squares minimization problem
min
xRn
||Axb||^22 (1)
is not unique. Lets consider a regularized form of (1):
min
xRn
(
||Axb||^22 +||x||^22
)
(2)
whereis a positive number. Let
(x) =||Axb||^22 +||x||^22.
(a) Find the gradient ofand use it to find problem (2)s equivalent of the normal
equations.
(b) Explain why problem (2) always has a unique solution. Hint: recall that symmetric
positive definite matrices are invertible.
(c) Consider a sequence of numbersn0 asn. Letxnbe the unique solution to
(2) with=nand letxbe the solution of (1) that has the smallest norm. Follow
the steps below to show thatxnxasn.
(i) Show||xn||^2 ||x||^2 for alln.
(ii) ShowATA(xnx)0 asn. Why can we not conclude from this alone
thatxnx0 asn?
(iii) ShowxnCol(AT) and use that to deducexnx0 asn.
Problem 4. Supposehis a known real-valued continuous function that has compact sup- port. When a signalfL 2 (0,1) is sent what is received is
[ 1 , 2 ,...,m] = [(Hf)(t 1 ),(Hf)(t 2 ),...,(Hf)(tm)]
3
whereHfis its convolution
(Hf)(t) =
1
0
h(ts)f(s)ds.
Consider the inverse problem of deter mining the signalffrom what is received. We cannot hope to reconstruct the whole signal from the finite information received so instead we look for the best approximation tofin thendimensional subspaceV ofL 2 (0,1) which has basis functions{ 1 , 2 ,…,n}(wheren < m).
(a) Formulate the problem of finding the best approximation tof inV in the least
squares sense as a problem in the form
min
Rn
||Ab||^2.
In other words, identify the matrixAand vectorb.
(b) If the matrixAdoes not have full rank, we consider the solution of the least squares
problem with minimum norm. express this solution in terms of the singular values
of the matrixA.
(c) If we define the row vector of functions Ln 2 (0,1) by = [ 1 , 2 ,...,n] then any
V can be written as= for some vectorRn. Show that
||||^2 L 2 (0,1)=TM
for some matrixMRnnand showMis positive definite and symmetric.
(d) Let >0 be given and consider the functionalJ:V Rdefined by
J() =||Ab||^2 +||||^2 L 2 (0,1)
where is the vector such that = . Formulate the problem of finding the
functionV that minimizesJas a least squares problem of the form
min
Rn
||A b||^2.
In other words, identify the matrixAand the vectorb.
(e) Verify that the least squares problem defined in part (e) has a unique solution and
find this solution.