Linear Algebra

Intro

Basic idea behind linear algebra: solving large number of linear algebraic equations simultaneously.

$a_{i,1} x_1 + a_{i,2} x_2 + a_{i,3} x_3 + \dots + a_{i,N} x_{N} = b_{N} \qquad i=1 \dots M$

There are N unknowns and M equations, forming a linear system from matrices:

$\mathbf{A \cdot x} = \mathbf{b}$

where lowercase letters denote vectors, and the dot represents multiplication of two matrices or a matrix and a vector, or dot product of two vectors.

Nonsingular vs Singular

We are not guaranteed to find a solution to a given linear system Ax=b

row degeneracy - one or more of the M equations is a linear combination of other equations
column degeneracy - one or more of the N variables are interchangeable and cannot be distinguished (are non-unique)
for square matrices, these two conditions are equivalent

Having singular system of equations means we can't find good solutions.

Tasks of Computational Linear Algebra

Beyond simply solving Ax=b for the unknowns x, we may want to do other things:

Solve more than one matrix equation, $\mathbf{A \cdot x}_j = \mathbf{b}_j$ , where the matrix A does not change but unknown/RHS vectors do change
Calculating the inverse of the matrix
Calculating the determinant of a square matrix
Calculating the condition number of a matrix

If M < N (row degeneracy), or if M = N and equations are degenerate, there are fewer equations than unknowns. If fewer equations than unknowns:

Either no solution vector, or
More than one solution vector
If more than one solution, solution space consists of particular solution added to linear combination of N-M vectors (which are in the nullspace of the matrix A)
SVD can be used to solve this problem, by find these vectors (SVD finds the nullspace of a matrix A)

If M > N (column degeneracy), there are fewer unknowns than equations. If fewer unknowns than equations:

Think least squares - more data points than unknowns
Looking for best fit/compromise solution
Linear least-squares problem can be used to solve overdetermined linear system

Linear Algebra Software

Software:

Blas is short for "basic linear algebra subroutines" and provides the linear algebra operations that compose Lapack
Lapack is the most common linear algebra package (Linpack is old and was replaced by Lapack)
ScaLapack is version of Lapack for parallel architectures
Lapack routines divided by precision, algorithm, simplifications (tridiag., sparse, dense, etc.)
Atlas is version of Lapack that automatically tunes itself to run as fast as possible for your system's architecture when it is compiled

Direct vs Iterative:

Major dividing line between routines
Direct - algorithms that execute in a predictable number of operations
Iterative - attempt to converge on the desired answer to within a desired tolerance, taking as many steps as necessary

Gauss-Jordan Elimination

Classic technique for solving linear equations: combine equations to cancel out unknowns, turning A into the identity matrix

Gauss-Jordan elimination produces the solution of the equations and the matrix inverse $\mathbf{A}^{-1}$ - this is an expensive extra thing to produce, not always desirable to have it

Deficiencies:

All right hand side values must be stored and manipulated at once (not feasible for very large systems of equations)
Gauss-Jordan is three times slower than alternative techniques that do not produce the matrix inverse

Advantages:

Straightforward
Solid
Pivoting

Pivoting:

Pivoting is just interchanging rows (partial pivoting) or rows and columns (full pivoting) to put desired elements on matrix diagonals
Partial pivoting is easier, full pivoting requires keeping track of permutations, and un-doing these when solution vector achieved

Storage requirements of method:

The matrix inverse of A is gradually built up in A, as original A is overwritten
Likewise solution vector x can gradually replace right-hand side vector b (solution vector is overwritten one row element at a time, each time an element is overwritten it will not be used again)

On the subject of row vs. column elimination strategy:

if we perform row operations, row transformations build right to left, so right hand side is transformed at each stage from one vector to another
if we perform column operations, the column operations build from left to right, so we have to remember each transformation at each stage, and apply them all in reverse order at the end
this means column operations are much less useful

Gaussian Elimination with Back-Substitution

Like Gauss-Jordan, Gaussian elimination utilizes pivoting to eliminate elements from the matrix A and simplify it, but doesn't go as far as simplifying it to the identity matrix - simplifies it to an upper-triangular matrix

Triangular matrices make back-substitution easy - solving first equation involves one unknown, solving second equation involves only one new unknown, etc.

Represents midpoint between full elimination (Gauss-Jordan) and triangular decomposition (LU, QR)

Advantage of Gaussian elimination with back-substitution over Gauss-Jordan elimination is faster in terms of number of operations:

Inner loops of Gauss-Jordan elimination have one subtraction and one multiplication, executed $$ N^3 $$ and $$ N^2 m $$ times (where m is number of right hand sides)
Inner loops in Gaussian elimination are only executed $\dfrac{1}{3} N^3$ times and $\dfrac{1}{2} N^2 m$ times
Back-substitution of RHS is $\dfrac{1}{2} N^2$ executions of similar loop (one mult plus one subt)
Gaussian elimination has a factor of three advantage over Gauss-Jordan, which erodes as m approaches N (number of RHS approaches number of equations/unknowns)

Calculation of inverse matrix is equivalent to the case of m = N (basically solving Ax=b, and letting b be each column of an identity matrix)

Gaussian elimination and back-subst. requires $\dfrac{1}{3} N^3$ operations for matrix reduction, $\dfrac{1}{2}N^3$ for RHS manipulation, and $\dfrac{1}{2} N^3$ for N back-substitutions, for a total of $\dfrac{4}{3} N^3$ operations
Gauss-Jordan only requires $$ N^3 $$ operations for matrix inversion, so it seems to win out
However, because all but one of RHS vector elements are 0, the operations count for Gaussian elimination is actually lower
Gauss-Jordan matrix inversion costs $\dfrac{1}{6} N^3$ operations

Note: the 1/6 comes from the fact that the matrix multiplication step normally requires 8 operations, but due to the sparseness of the RHS, we are only performing 1 of the 8 operations. Hence, $\dfrac{1}{8} \times \dfrac{4}{3} = \dfrac{1}{6}$

LU Decomposition

Main article: LU Decomposition

This is based on decomposing the matrix A into two matrices, L and U:

$\mathbf{A} = \mathbf{L \cdot U}$

L is a lower triangular matrix, U is an upper triangular matrix.

To solve the linear system:

$\mathbf{A \cdot x} = \mathbf{b}$

we proceed in two steps:

Equation 1 is solved by forward substitution:

$\mathbf{L \cdot y} = \mathbf{b}$

and the second equation is solved by back substitution:

$\mathbf{U \cdot x} = \mathbf{y}$

The advantage is that solving a linear system with a triangular matrix is much easier.

The back and forward substitutions execute $$ N^2 $$ operations. If there are N RHS vectors (i.e., if we're explicitly inverting a matrix), and we take into account the reduced number of operations due to all but one element in each RHS being zero, the total number of operations is:

For forward substitution, the operation count is: $\dfrac{1}{6} N^3$

For back substitution, the operation count is: $\dfrac{1}{2} N^3$

BUT, there is also the cost of performing the decomposition itself, which is not free. This can be done with Crout's Algorithm.

Adding the cost of the LU decomposition itself brings the total cost of inverting a matrix to $$ N^3 $$ , which is the same complexity class as Gaussian elimination. This highlights an important fact: LU decomposition is simply a modified version of Gaussian elimination.

The advantage of LU decomposition is that, once A has been decomposed into L and U, it remains unchanged and can be used to solve as many RHS as desired, one at a time. (Gauss-Jordan and Gaussian elimination both depend on manipulating the left and right sides simultaneously.)

Tridiagonal and Band-Diagonal Systems

Tridiagonal matrix systems have only two non-zero off-diagonal elements. This structure commonly arises from finite difference schemes.

For tridiagonal matrices, LU decomposition forward and back substitution only takes $$ O(N) $$ operations, instead of the usual $$ O(N^2+N) $$

To store tridiagonal systems, it is wasteful to store the entire matrix, so only three vectors are used. The algorithm to solve the system performs decomposition and forward subsittution for each element, then performs back-substitution. Pivoting is not necessary, but this means the algorithm will failif there are zeros on the diagonals. However, many applications that lead to tridiagonal matrices have guarantees that will prevent this from failing.

Diagonal dominance: property that guarantees a zero pivot will not be encountered. This condition states:

$| b_j | > | a_j | + | c_j | \qquad j = 0, \dots, N-1$

where j indexes a row, the b elements are on the diagonal, and a and c are the off-diagonal elements.

Tridiagonal algorithm is more robust in practice than in theory - a rare occurrence.

Parallel Solution

To solve tridiagonal systems in parallel, the (large) matrix system can be permuted, and columns combined, to form a smaller tridiagonal system in the bottom-right part of the matrix.

The process starts by permuting the columns of A (and the solution vector) so that the vector (u0, u1, u2, u3, u4, u5, u6) would become the vector (u0, u2, u4, u6, u1, u3, u5). This leads to a banded diagonal matrix - a matrix with a single diagonal, plus two additional off-diagonal bands above and two off-diagonal bands below that diagonal.

Next, the even and odd rows are combined to eliminate nonzero elements in the lower left, which turns the bottom right corner of the matrix into a tridiagonal matrix.

Now the original tridiagonal linear system of size N has been reduced to a smaller tridiagonal system of size floor(N/2).

Visually:

Band Diagonal Systems

Tridiagonal systems have nonzero elements on the diagonal plus or minus one. Band diagonal systems are more general, and have m1 >= 0 nonzero elements below the diagonal, m2 >= 0 nonzero elements above the diagonal.

These linear systems can be solved using LU decomposition very fast with less storage than N x N systems.

Band-diagonal matrices: stored in compact form (matrix tilted 45 degrees clockwise, forms a block matrix of size (m1 + 1 + m2) x N, with a few nonzero elements at the ends. Algorithms should be tailored to the storage format.

It is not possible to store LU decomposition of band-diagonal matrix as compactly as the compact form of A. Crout's method produces additional nonzero fill-ins. Can store upper triangular factor U in a matrix with same shape as A, and store lower triangular matrix L in separate compact matrix of size N x m1. Diagonal elements of U are in first column of U.

Once matrix A is decomposed, the system can be solved for any number of RHS vectors and solved via back-substitution, as standard with LU systems.

iterative Improvement

Iterative improvement is the concept of using a (slightly) wrong solution to the matrix equation $\mathbf{A \cdot x} = \mathbf{b}$ to compute a residual, and use the residual to improve the solution.

We are attempting to solve the equation $\mathbf{A} \cdot \mathbf{x} = \mathbf{b}$ .

Our slightly wrong solution solves the equation $\mathbf{A} \cdot \left( \mathbf{x} + \delta \mathbf{x} \right) = \left( \mathbf{b} + \delta \mathbf{b} \right)$ .

Subtracting the first from the second gives $\mathbf{A} \cdot \delta \mathbf{x} = \delta \mathbf{b}$ .

Now we can obtain the portion that our solution is off by, $\delta \mathbf{x}$ , through the equation:

$\mathbf{A} \cdot \delta \mathbf{x} = \mathbf{A} \cdot \left( \mathbf{x} + \delta \mathbf{x} \right) - \mathbf{b}$

Finally, once we have the error $\delta \mathbf{x}$ , we subtract it from our slightly wrong solution $\mathbf{x} + \delta \mathbf{x}$ to obtain an improved solution. (And this procedure can be repeated again, and again, until our numerical precision runs out.)

If we decompose A using LU decomposition, we get an extra computational savings from being able to re-use it for each step of the solution. Iterative improvement can be implemented as a function in an LU decomposition class, since it makes use of LU decomposition.

Iterative improvement is highly recommended as a follow-up step to solving the original matrix equation Ax = b. iterative improvement is $$ O(N^2) $$ (due to a vector-matrix multiply, and a forward and back substitution to solve an LU decomposed system). This is cheap relative to the $$ O(N^3) $$ operations for solving the original matrix equation.

Iterative improvement applied repeatedly can be looked at as an iterative Newton-Raphson method for root-finding, applied to matrix inversion.

Singular Value Decomposition

Main article: SVD Decomposition

SVD:

Powerful set of techniques for dealing with sets of equations that are singular or numerically near-singular
Can diagnose problems with matrix systems where LU or Gaussian elimination fails.
Method of choice for solving linear least-squares problems

Basic theorem of linear algebra: any M x N matrix A can be written as product of three matrices $\mathbf{U} \cdot \mathbf{W} \cdot \mathbf{V}^{T}$ :

U is an M x N column-orthogonal matrix
W is an N x N diagonal matrix with positive or zero elements (singular values)
V is the transpose of an N x N orthogonal matrix

M > N corresponds to the overdetermined situation (more equations than unknowns)

M < N corresponds to the underdetermined situation (fewer equations than unknowns)

Decomposition into $\mathbf{U} \cdot \mathbf{W} \cdot \mathbf{V}^{T}$ can always be done

W contains the singular values - it is conventional to return them in sorted, descending order

Sparse Linear Systems