Mathematics, 03.03.2020 01:01 qwertylol12345

3.10. This problem considers the question, why not use UL instead of LU? You can assume that A is a 2 ×2 matrix. (a) Find a Doolittle version of the factorization A = UL, where L is a unit lower triangular matrix and U is upper triangular. You can assume pivoting is not necessary. (b) Describe the resulting algorithm for solving Ax = b. (c) Is there any connection between the Doolittle factorization of AT and the one you found in part (a)?

Answers: 3

Show answers

Answers

Answer from: rfdgd

The Doolittle version of factorization A =UL is

$\left[\begin{array}{ccc}a_{11}&a_{12\\a_{21}&a_{22}\end{array}\right] =\left[\begin{array}{ccc}a_{11}-a_{12}*\frac{a_{21}}{a_{22}} &a_{12}\\0&a_{22}\end{array}\right]\left[\begin{array}{ccc}1&0\\\frac{a_{21}}{a_{22}} &1\end{array}\right]$

Algorithm

1) Obtain the values of the individual elements of U and L

$U_{22} = a_{22}$

$l_{21} = \frac{a_[21} {U_{22}}}$

$U_{12} = a_{12}$

$U_{11} = a_{11} - U_{12}*l_{21}$

2) Obtain the value of z for the expression Uz = b 1.e

$\left[\begin{array}{ccc}U_{11}&U_{12\\0&U_{22}\end{array}\right] \left[\begin{array}{ccc}z_1\\z_2\end{array}\right] =\left[\begin{array}{ccc}b_1\\b_2\end{array}\right]$

$z_2 =\frac{b_2}{U_22}$

$z_1 = \frac{1}{U_[11}} [b_1 - U_{12}\ z_2]$

3 Obtain the value of x for the expression Lx = z 1.e

$\left[\begin{array}{ccc}1&0\\l_{21}&1\end{array}\right] \left[\begin{array}{ccc}x_1\\x_2\end{array}\right] = \left[\begin{array}{ccc}z_1\\z_2\end{array}\right]$

x_1 = z_1

$x_2 = z_2 -l_{21}x_1$

Let say that A^T =LU is the Doolittle factorization of A^T

where L is a unit lower triangular matrix and U is upper triangular matrix

Hence A = U^T L^T

Here L^T is a unit upper triangular matrix and U^T lower is triangular

In the question that we are given A = UL meaning that A^T = L^TU^T looking at this U is upper triangular matrix and L is unit lower triangular matrix. So we can say that the Doolittle version given behave like Doolittle factorization only for A and there is no connection with A^T

Step-by-step explanation:

From the question let assume that $A =\left[\begin{array}{ccc}a_{11}&a_{12\\a_{21}&a_{22}\end{array}\right]$

The objective is to find the Doolittle version of the factorized A = UL and this can be expressed in matrix as

$\left[\begin{array}{ccc}a_{11}&a_{12\\a_{21}&a_{22}\end{array}\right] =\left[\begin{array}{ccc}U_{11}&U_{12\\0&U_{22}\end{array}\right]\left[\begin{array}{ccc}1&0\\l_{21}&1\end{array}\right]$

Looking at this in form of matrix multiplication

$a_{11} =U_{11} +U_{12} l_{21}$

$a_{12} = U_{12}$

$a_{21} = U_{22} l_{21}$

$a_{22} = U_{22}$

=> $l_{21} = \frac{a_{21}}{U_{22}} = \frac{a_{21}}{a_{22}}$

and $u_{11} = a_{11} - U_{12}l_{21} = a_{11} -a_{12} *\frac{a_{21}}{a_{22}}$

Hence $U = \left[\begin{array}{ccc}a_{11}-a_{12}*\frac{a_{21}}{a_{22}} &a_{12}\\0&a_{22}\end{array}\right]$

$L = \left[\begin{array}{ccc}1&0\\\frac{a_{21}}{a_{22}} &1\end{array}\right]$

b) From the question we are given

Ax =b

And objective is to describe the algorithm to solve it

Ax =b

UL x = b

Let assume that z = Lx

U z = b

Given that U is upper triangular matrix then y can be obtained by back substitution

Now that we are able to obtain y, given that L is a unit lower triangular matrix we can obtain x by forward substitution

The algorithm is given in the answer section

Answer from: Quest

the distance between point a and point b is 6 units

What is the distance between points a and b?

Answer from: Quest

theres no picture: /

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 16:00

On saturday morning, calls arrive at ticketmaster at a rate of 108 calls per hour. what is the probability of fewer than three calls in a randomly chosen minute?

Answers: 1

Answer

Mathematics, 21.06.2019 18:30

In the following diagram it is given that dec,ab || dc, ad || eb, and ad is congruent to bc. a)why isn't abc a parallelogram even though it has a pair of parallel sides and a pair of congruent sides b)explain why be must be congruent to bc. further explain what type of triangle this makes triangle abc and what it tells you about angle 1 and angle 2c) finally why must angle 3 be congruent to angle 1? further, explain why we know that angle 3 is congruent to angle 2

Answers: 1

Answer

Mathematics, 21.06.2019 19:00

Explain why the factor 1.5 is broke. into two numbers in the model.

Answers: 3

Answer

Mathematics, 21.06.2019 19:20

Thompson and thompson is a steel bolts manufacturing company. their current steel bolts have a mean diameter of 127 millimeters, and a variance of 36. if a random sample of 35 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by greater than 0.5 millimeters? round your answer to four decimal places.

Answers: 3

Answer

You know the right answer?

3.10. This problem considers the question, why not use UL instead of LU? You can assume that A is a...

Questions