linear algebra
math | linear algebra – 这个题目属于一个 linear algebra的代写任务, 是比较有代表性的 linear algebra等代写方向
Show that the sequencexn= 312 n(n= 1, 2 ,…) converges quadratically asn.
Question 2(15 marks in total). Consider the fixed point iteration scheme pn+1=g(pn).
(2-a) (5 marks). State the fixed point iteration theorem, i.e., the theory for ensuring convergence of the above scheme to fixed pointpin the interval [a,b].
(2-b) (5 marks). Describe the upper bound for the absolute error|pn|.
(2-c) (5 marks). It is known that ifg(x)Ck([a,b]) fork2 andg([a,b])[a,b]
g(p) ==g(k1)(p) = 0,
thenxnconverges to a fixed point withkth order of convergence. Consider the following iteration for calculating a fixed point^1 /^3 :
xn+1=axn+b
x^2 n
+c
^2
x^5 n
Assuming that this iterative scheme converges asx 0 sufficiently close to^1 /^3 , determine the values ofa,b,csuch that the method has cubic convergence rate.
Considery= sinx, (3-a) (5 marks). Obtain the Lagrange interpolating polynomial from the data
sin 0 = 0, sin
6
=
1
2
, sin
3
=
3
2
, sin
2
= 1
and use it to evaluate sin 12 . (keep 4 significant digits)
(3-b) (5 marks). Find the error bound of the Lagrange interpolating polynomial atx= 12 . Compare the error bound with the actual error at the pointx= 12 .
(3-c) (10 marks). LetPnbe the degreenLagrange interpolating polynomial of cos(2x) on the uniformly spaced nodesx 0 ,,xnon [0,1] withxj=jh,h= 1/n. prove that
max
0 x 1
|cos(2x)Pn(x)| 0 asn.
(4-a) (5 marks). Use the most accurate three-point endpoint formula to determine the missing valuef(1.4) in the following table (keep 4 significant digits):
x f(x) f(x)
1. 1 9. 025
1. 2 11. 02
1. 3 13. 46
1. 4 16. 44
(4-b) (5 marks). The data in part (4-a) were taken from the functionf(x) =e^2 x. Compute the actual error atx= 1.4 and the error bound using the error formula.
Question 5(15 marks). Consider the definite integral
3
1 ln
(
x^2
)
dx. (5-a) (5 marks). Approximate this integral using the composite Simpsons rule with 5 equal spaced nodesx 0 , x 1 ,,x 4.
(5-b) (5 marks). Give a rigorous theoretical upper bound on the absolute error in your answer to (5-a).
(5-c) (5 marks). Based on the upper bound obtained in (5-b), determine theoretical values ofnandh required to obtain an approximation accurate to within 10^4 using the composite Simpsons rule.
Given the initial-value problem
y=
2
t
y+t^2 et, 1 t 2 , y(1) = 0,
with exact solutiony(t) =t^2 (ete) : (6-a) (5 marks). Use Eulers method withh= 0.1 to approximate the values ofyatt= 1.5 andt= 1.6.
(6-b) (5 marks). Use the answers generated in part (6-a) and linear interpolation to approximatey(1.55).
(6-c) (5 marks). Based on the theoretical error estimate, compute the largest value ofhto ensure that |yii| 0 .5 (the answer should be in the form x.xxxx).
Let
A=
2 4 6
4 10 16
6 16 28
be a given matrix inR^3 ^3. (7-a) (5 marks). Find a lower tri angular matrixLand a diagonal matrixDsuch thatA=LDL>.
(7-b) (5 marks). Compute an approximation to the largest eigenvalue ofAby performing two steps of the power method withx 0 = [1, 1 ,1]T. Keep at least 4 decimal places in calculations.
Consider the following linear system:
x 1 +x 2 + 3x 3 = 5 x 1 + 3x 2 +x 3 = 2 3 x 1 +x 2 +x 3 = 4 (8-a) (5 marks). Re-order the equations so that Jacobi iteration will converge to the exact solution as fast as possible. Carry out 2 iterations with starting vectorx(0)= (0, 0 ,0). Explain why the iteration is convergent.
(8-b) (5 marks). The spectral radius of a matrix is difficult (expensive) to calculate. Fortunately, for any matrixMwe have (by a theorem of linear algebra) that(M) Mwhere(M) is the spectral radius and is any matrix norm. Write your Jacobi iteration from (a) in the formx(n+1)=Tx(n)+b. CalculateT and use this value to estimate the error after 20 Jacobi iterations.