Mathematics: 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)?

3.10. This problem considers the question, why

Ответ:

gunruner21

20.01.2022

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

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