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Chapter 4: Problem 31

Consider $$ A=\left[\begin{array}{rrr} 3 & 2 & -1 \\ 6 & 6 & 2 \\ -1 & 1 & 3 \end{array}\right] $$ Use Gaussian elimination with scaled row pivoting to obtain the factorization $$ P A=L D U $$ where \(L\) is a unit lower triangular matrix, \(U\) is a unit upper triangular matrix, \(D\) is a diagonal matrix, and \(P\) is a permutation matrix.

Short Answer

Expert verified
Perform Gaussian elimination with row exchanges to find \(L\), \(D\), \(U\), and \(P\). Verify with multiplication: \(PA = LDU\).

Step by step solution

01

Initialize Matrices

First, we must prepare the matrices we will eventually construct: the permutation matrix \(P\), the lower triangular matrix \(L\), the diagonal matrix \(D\), and the upper triangular matrix \(U\). Initially, set \(P\) and \(L\) as identity matrices of order 3, and \(U\) equal to the given matrix \(A\). Likewise, initialize \(D\) as an identity matrix.
02

Determine the Scaling Factors

Calculate the scaling factors for each row. The scaling factor is the largest absolute value of the elements in each row of \(U\). For the matrix \(A\), the scaling factors are as follows: Row 1: 3, Row 2: 6, Row 3: 3.
03

Scaled Partial Pivoting for Row 1

We choose a pivot row by comparing the ratio of the first element of each row to its scaling factor. For Row 1: \(|3|/3 = 1\). For Row 2: \(|6|/6 = 1\). For Row 3: \(|-1|/3 \approx 0.33\). Since Row 1 and Row 2 have the same ratio, we choose Row 2 based on its higher index (larger absolute value in case of a tie). Interchange Row 1 and Row 2 in both \(P\) and \(U\). Update the permutation matrix \(P\).
04

Elimination for Column 1

Use Row 1 to eliminate elements below the pivot position in Column 1 of \(U\). After row exchanges, Row 1 is \([6, 6, 2]\) and Row 2 is \([3, 2, -1]\). For Row 3, we use the operation: NewRow2 = Row2 - (3/6)*Row1 = \([3, 2, -1] - 0.5*[6, 6, 2]\). Perform the subtraction to get the new Row 2. Update the matrix \(L\) with 0.5 in position \(L_{21}\).
05

Adjust Diagonal Matrix D

Now, factor the diagonal elements. For the first column's pivoting step, the factor is 6. Set \(D_{11} = 6\) and divide the first row of \(U\) by this factor to maintain the unit upper triangular structure: \([1, 1, \frac{1}{3}]\). Update \(U\).
06

Scaled Partial Pivoting for Row 2

Find the pivot for Row 2 in the submatrix starting from the second column. Calculate as done previously but only for Row 2 and Row 3 now (after previously updated matrix). Interchange if necessary, adjust \(P\), \(L\), and \(U\) based on changes.
07

Elimination and Adjust D for Column 2

Perform elimination on Row 3 using Row 2 to zero out the second element in Row 3. Then update \(D\) with the new pivot found in Step 6, and adjust \(U\) to maintain the unit upper triangular form as before.
08

Resulting Matrices

After repeating the steps similarly for the third column, all non-diagonal entries below the main diagonal must be zero in \(U\), \(D\) must be a diagonal matrix with previous pivot entries, and \(L\) and \(P\) must reflect row exchanges. Verify by multiplying \(P A = L D U\) with the final matrices. Adjust any small numerical inaccuracies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Factorization
Matrix factorization, often encountered in linear algebra, involves breaking down a matrix into a product of multiple matrices. This process can simplify many computations, especially in solving linear equations or finding matrix inverses. In the context of Gaussian elimination, we aim to decompose matrix \( A \) into matrices \( P \), \( L \), \( D \), and \( U \).
These matrices have distinct roles:
  • \( P \): a permutation matrix used to reorder rows.
  • \( L \): a unit lower triangular matrix that holds multipliers used in the elimination process.
  • \( D \): a diagonal matrix representing scaling factors ensuring numerical stability.
  • \( U \): a unit upper triangular matrix resulting from transformations applied to \( A \).
By breaking \( A \) into \( P \times L \times D \times U \), we facilitate easier computations for various matrix operations.
Scaled Row Pivoting
Scaled row pivoting is a technique used in Gaussian elimination to combat numerical instability and ensure accurate results. This approach involves selecting pivot elements in a way that prevents large round-off errors. Here's how it works:
  • First, calculate scaling factors for each row. These are the largest absolute values of elements in each row.
  • Determine pivot candidates by comparing ratios of each row's leading entry to its scaling factor.
  • Select the row with the highest ratio as the pivot row. This reduces the likelihood of small divisors, minimizing error propagation.
For example, when comparing rows, a row with a ratio of 1 has a better scaling potential than one with a ratio of 0.5. Such scaled pivoting ensures the algorithm remains stable and reliable, especially for equations involving small and large numbers.
Permutation Matrix
A permutation matrix is a fundamental tool in matrix factorization, particularly when dealing with row exchanges. In Gaussian elimination, it's crucial to ensure the correct order of operations by dynamically adjusting rows.
  • Permutation matrix \( P \) rearranges rows of another matrix according to pivoting decisions.
  • \( P \) is an identity matrix that undergoes row swaps during factorization to record row interchanges.
  • Each row swap within the original matrix \( A \) is mirrored by swapping rows within \( P \).
This matrix helps to establish a clear track of any row reordering done during elimination, ensuring matrices align correctly in the factorized form \( P \times L \times D \times U \). Understanding how \( P \) functions provide insight into the orderly execution of row operations.

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Most popular questions from this chapter

For any \(n \times n\) matrix \(A\), define $$ \|A\|_{F}=\left(\sum_{i=1}^{n} \sum_{j=1}^{n} a_{i j}^{2}\right)^{1 / 2} $$ (This is called the Frobenius norm.) Is this a subordinate matrix norm? Answer the same question for \(\|A\|=\max _{1 \leq i, j \leq n}\left|a_{i j}\right| .\) Prove that these equations define norms on the vector space of all \(n \times n\) matrices.

This problem shows how the solution of a system of equations can be unstable relative to perturbations in the data. Solve \(A x=b\), with \(b=(100,1)^{T}\) and with each of the following matrices. (See Stoer and Bulirsch [1980, p. 185].) $$ A_{1}=\left[\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right] \quad A_{2}=\left[\begin{array}{rr} 1 & 1 \\ 1 & 0.01 \end{array}\right] $$

Consider the linear system $$ \left[\begin{array}{rrr} 2 & 0 & -1 \\ -2 & -10 & 0 \\ -1 & -1 & 4 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]=\left[\begin{array}{r} 1 \\ -12 \\ 2 \end{array}\right] $$ Using the starting vector \(x^{(0)}=(0,0,0)^{T}\), carry out two iterations of each of the following methods: a. Jacobi b. Gauss-Seidel c. Conjugate gradient

Let \(A\) be diagonally dominant, and let \(Q\) be the lower triangular part of \(A\), as in the Gauss-Seidel method. Prove that \(\rho\left(I-Q^{-1} A\right)\) is no greater than the largest of the ratios $$ r_{i}=\left\\{\sum_{j=i+1}^{n}\left|a_{i j}\right|\right\\} /\left\\{\left|a_{i i}\right|-\sum_{j=1}^{i-1}\left|a_{i j}\right|\right\\} $$

Prove that if \(A\) is ill conditioned, then a singular matrix exists within distance \(\|A\|_{2} / \kappa(A)\) of \(A\).
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