Solve the following linear systems by the Conjugate Gradient...

Solve the following linear systems by the Conjugate Gradient Method, keeping track of the residual vectors and solution approximations as you iterate. (a) $$ \begin{gathered} \left(\begin{array}{rr} 3 & -1 \\ -1 & 5 \end{array}\right) \mathbf{u}=\left(\begin{array}{l} 2 \\ 1 \end{array}\right), \quad(b)\left(\begin{array}{rrr} 6 & 2 & 1 \\ 2 & 3 & -1 \\ 1 & -1 & 2 \end{array}\right) \mathbf{u}=\left(\begin{array}{r} 1 \\ 0 \\ -2 \end{array}\right), \quad(c)\left(\begin{array}{rrr} 6 & -1 & -3 \\ -1 & 7 & 4 \\ -3 & 4 & 9 \end{array}\right) \mathbf{u}=\left(\begin{array}{r} -1 \\ -2 \\ 7 \end{array}\right), \\ \text { (d) }\left(\begin{array}{rrrr} 6 & -1 & -1 & 5 \\ -1 & 7 & 1 & -1 \\ -1 & 1 & 3 & -3 \\ 5 & -1 & -3 & 6 \end{array}\right) \mathbf{u}=\left(\begin{array}{r} 1 \\ 2 \\ 0 \\ -1 \end{array}\right), \quad(e)\left(\begin{array}{llll} 5 & 1 & 1 & 1 \\ 1 & 5 & 1 & 1 \\ 1 & 1 & 5 & 1 \\ 1 & 1 & 1 & 5 \end{array}\right) \mathbf{u}=\left(\begin{array}{l} 4 \\ 0 \\ 0 \\ 0 \end{array}\right) . \end{gathered} $$

Solve the following linear systems by the Conjugate Gradient Method, keeping track of the residual vectors and solution approximations as you iterate.
(a)
$$
\begin{gathered}
\left(\begin{array}{rr}
3 & -1 \\
-1 & 5
\end{array}\right) \mathbf{u}=\left(\begin{array}{l}
2 \\
1
\end{array}\right), \quad(b)\left(\begin{array}{rrr}
6 & 2 & 1 \\
2 & 3 & -1 \\
1 & -1 & 2
\end{array}\right) \mathbf{u}=\left(\begin{array}{r}
1 \\
0 \\
-2
\end{array}\right), \quad(c)\left(\begin{array}{rrr}
6 & -1 & -3 \\
-1 & 7 & 4 \\
-3 & 4 & 9
\end{array}\right) \mathbf{u}=\left(\begin{array}{r}
-1 \\
-2 \\
7
\end{array}\right), \\
\text { (d) }\left(\begin{array}{rrrr}
6 & -1 & -1 & 5 \\
-1 & 7 & 1 & -1 \\
-1 & 1 & 3 & -3 \\
5 & -1 & -3 & 6
\end{array}\right) \mathbf{u}=\left(\begin{array}{r}
1 \\
2 \\
0 \\
-1
\end{array}\right), \quad(e)\left(\begin{array}{llll}
5 & 1 & 1 & 1 \\
1 & 5 & 1 & 1 \\
1 & 1 & 5 & 1 \\
1 & 1 & 1 & 5
\end{array}\right) \mathbf{u}=\left(\begin{array}{l}
4 \\
0 \\
0 \\
0
\end{array}\right) .
\end{gathered}
$$

Key Concepts

Iterative Methods

Iterative methods are techniques used to obtain approximate solutions to problems by successive refinement, rather than computing the exact solution in one step. Such methods are especially useful for large-scale linear systems where direct solution methods become impractical due to their computational and memory requirements. The Conjugate Gradient Method is an example, incrementally improving the solution with each iteration.

Residual Vectors

Residual vectors are the differences between the right-hand side of the linear system and the result of applying the coefficient matrix to the current approximated solution. They measure the current error in the solution and are critical in iterative methods like the Conjugate Gradient Method, where they guide the selection of the next search direction and help in assessing convergence of the approximation.

Solution Approximations

Solution approximations refer to the iterative estimates of the true solution obtained during the execution of an iterative method. In the Conjugate Gradient Method, each step produces an updated solution approximation that ideally reduces the error, as indicated by the diminishing magnitude of the residual vector. Monitoring these approximations is important for understanding the convergence behavior of the method.

Conjugate Gradient Method

The Conjugate Gradient Method is an iterative algorithm specifically designed for solving systems of linear equations whose coefficient matrix is symmetric and positive definite. It works by generating a sequence of improving approximate solutions and corresponding residuals, using the concept of conjugate (or A-orthogonal) directions which ensures that the error is minimized along these directions. This method is particularly efficient for large, sparse systems where direct methods would be computationally expensive.

Symmetric Positive Definite Matrices

Symmetric positive definite matrices are those that are both symmetric (A = A?) and have all positive eigenvalues, ensuring that the quadratic form x?Ax is strictly positive for any nonzero vector x. This property guarantees the convergence of the Conjugate Gradient Method, as it underpins the algorithm’s ability to define a valid inner product induced by the matrix and to construct mutually conjugate search directions.

Transcript

00:01 Okay, so if a is a matrix of order, let's say, m by n, and x is a column vector of order n by one, then the product of a times x can be expressed as a linear combination of the column vectors of a in which the coefficients are the entries of x.

00:20 So given here, we have a matrix a, so we have in the form, a, x is equal to b, right? well, we can factor out an x1, x2, and x3.

00:34 So we have x1 times our first column vector of 5 negative 1 ,0, plus x2 times our next column vector, plus x3 times our next column vector is equal to, well, the given column vector here.

00:47 That implies that 5x1 minus 1, so minus x, minus x, minus 1 times x1, so minus x1.

01:01 And then we have, well, times zero times x1.

01:06 So zero x1.

01:15 Plus 6x2, minus 2x2.

01:22 And then we have 4x2.

01:28 And then plus negative 7x3.

01:33 And then plus 3x4.

01:35 And a negative, well, negative 1 x3.

01:39 So negative x3 is equal to the given column vector of 203.

01:50 Okay.

01:54 So while we get here, we get 5x1 minus, or 5x1 plus 6x2 minus x3 is equal to 2.

02:04 And then, so, yeah, we get 5x1 plus 6x2.