Lecture 13
The advent of high speed computers and sophisticated software tools have
made the computation of derivatives for functions defined by evaluation
programs both easier and more important.
On one hand, the dependence of certain program outputs with respect to
certain input parameters can now be determined
and quantified more or less automatically, i.e. without the user having to
append or rewrite the function evaluation procedure.
On the other hand, such qualitative and quantitative dependence analysis is
invaluable for the optimization of key output objectives with respect to
suitable decision variables or the identification of model parameters with
respect to given data.
In fact, we may juxtapose the mere simulation of a physical or social system
by repeatedly running an appropriate computer model for various input data
to its optimization by a systematic adjustment of certain decision variables
and model parameters.
The transition from the former computational paradigm to the latter may be
viewed as a central characteristic of present day scientific computing.
In the coming lectures we will first address the issue of calculating
derivatives using finite difference approximations for approximating:
1. Gradient
2. Sparse Jacobian
3. Approximating the Hessian ( matrix of second order derivatives)
4. Approximating a sparse Hessian
And then we will focus our attention on automatic differentiation using
mainly the book of Andreas Griewank and material of TAMC as well as
other materials.
Finite differencing
Based on Taylor’s theorem following change in function values as a
response to small perturbations of unknowns , we estimate the response to
infinitesimal perturbations namely the derivatives
The partial derivative of a smooth function
f:
n

with respect to the i-th variable I , may be approximated by a central
difference formula:
f ( x   ei )  f ( x   ei )
f

 error
xi
2
Here  is a small positive scalar while ei is the i-th unit vector that means
that all the elements of this vector are 0 except for a 1 in the i-th position.
Finite Difference Derivative Approximation
Finite differencing is an approach to the calculation of approximate
derivatives whose motivation (like that of so many algorithms in
optimization) comes from Taylor’s theorem.
Many software packages perform automatic calculation of finite differences
whenever the user is unable or unwilling to supply a code to calculate exact
derivatives.
Although they yield only approximate values for the derivatives, the results
are adequate in many situations.
By definition, derivatives are a measure of the sensitivity of the function to
infinitesimal changes in the values of the variables.
Our approach in this section is to make small, finite perturbations in the
values of x and examine the resulting differences in the function values.
Taking ratios of the function difference to variable difference we are able
to obtain approximations to the derivatives.
Gradient approximation
An approximation to the gradient vector f ( x) can be obtained by
evaluating the function f at (n + 1) points and performing some elementary
arithmetic.
We describe this
technique, along with amore accurate variant that requires additional
function evaluations.
A popular formula for approximating the partial derivative ∂ f/∂xi at a given
point x is the forward-difference, or one-sided-difference, approximation,
defined as
(8.1)
The gradient can be built up by simply applying this formula for i _ 1, 2, . . .
, n.
This process requires evaluation of f at the point x as well as the n perturbed
points x +
 ei ,
i =1, 2, . . . , n: a total of (n + 1) points.
The basis for the formula (8.1) is Taylor’s theorem, Theorem 2.1 in Chapter
2 of the book of Nocedal and Wright, second Edition.
When f is twice continuously differentiable, we have:
If we choose L to be a bound on the size of  2 f (.) in the region of interest,
it follows directly from this formula that the last term in this expression is
bounded by
2
(L/2 p ), so that
We now choose the vector p to be
 ei , so that it represents a small
change in the value
of a single component of x (the i-th component).
For this p, we have that  f (x)T p =  f (x)T ei = ∂ f/∂xi , so by rearranging
(8.3), we conclude that :
(8.4)
We derive
term  
the forward-difference formula (8.1) by simply ignoring the error
in this
expression, which becomes smaller and smaller as  approaches zero.
An important issue in implementing the formula (8.1) is the choice of the
parameter  . The error expression (8.4) suggests that we should choose 
as small as possible. Unfortunately,
this expression ignores the round off errors that are introduced when the
function f is evaluated on a real computer, in floating-point arithmetic. From
our discussion in the
Appendix (see (A.30) and (A.31)), we know that the quantity u known as
unit roundoff is crucial: It is a bound on the relative error that is introduced
whenever an arithmetic operation is performed on two floating-point
numbers. (u is about 1.1 × 10−16 in double precision
IEEE floating-point arithmetic.)
The effect of these errors on the final computed
value of f depends on the way in which f is computed. It could come from an
arithmetic formula, or from a differential equation solver, with or without
refinement.
As a rough estimate, let us assume simply that the relative error in the
computed f is bounded by u, so that the computed values of f (x)
and f (x +  ei ) are related to the exact values in the following way:
where comp(·) denotes the computed value, and L f is a bound on the value
of | f (·)| in the region of interest. If we use these computed values of f in
place of the exact values in (8.4) and (8.1), we obtain an error that is
bounded by:
ERROR ANALYSIS AND FLOATING-POINT ARITHMETIC
In most of this book our algorithms and analysis deal with real numbers.
Modern digital computers, however, cannot store or compute with general
real numbers. Instead they work with a subset known as floating-point
numbers. Any quantities that are stored on the computer, whether they are
read directly from a file or program or arise as the
intermediate result of a computation, must be approximated by a floatingpoint number.
In general, then, the numbers that are produced by practical computation
differ from those that would be produced if the arithmetic were exact.
Of course, we try to perform our computations in such a way that these
differences are as tiny as possible.
Discussion of errors requires us to distinguish between absolute error and
relative error.
If x is some exact quantity (scalar, vector, matrix) and x is its approximate
value, the
absolute error is the norm of the difference, namely, x  x . (In general, any
of the norms
(A.2a), (A.2b), and (A.2c) ( See Nocedal and Wright Appendix A ) can be
used in this definition.) The relative error is the ratio of
the absolute error to the size of the exact quantity, that is,
Definition of sparse matrix
In the mathematical subfield of numerical analysis a sparse matrix is a
matrix populated primarily with zeros.
Conceptually, sparsity corresponds to systems which are loosely coupled.
Consider a line of balls connected by springs from one to the next; this is a
sparse system. By contrast, if the same line of balls had springs connecting
every ball to every other ball, the system would be represented by a dense
matrix. The concept of sparsity is useful in combinatorics and application
areas such as network theory, of a low density of significant data or
connections.
Huge sparse matrices often appear in science or engineering when solving
partial differential equations.
When storing and manipulating sparse matrices on a computer, it is
beneficial and often necessary to use specialized algorithms and data
structures that take advantage of the sparse structure of the matrix.
Operations using standard matrix structures and algorithms are slow and
consume large amounts of memory when applied to large sparse matrices.
Sparse data is by nature easily compressed, and this compression almost
always results in significantly less memory usage. Indeed, some very large
sparse matrices are impossible to manipulate with the standard algorithms.
Figure Illustration of Sparse Matrix
Storing a sparse matrix
The naive data structure for a matrix is a two-dimensional array. Each entry in the array
represents an element ai,j of the matrix and can be accessed by the two indices i and j. For
a m×n matrix we need at least enough memory to store (m×n) entries to represent the
matrix.
Many if not most entries of a sparse matrix are zeros. The basic idea when storing sparse
matrices is to store only the non-zero entries as opposed to storing all entries. Depending
on the number and distribution of the non-zero entries, different data structures can be
used and yield huge savings in memory when compared to a naïve approach.
One example of such a sparse matrix format is the (old) Yale Sparse Matrix Format[1]. It
stores an initial sparse m×n matrix, M, in row form using three one-dimensional arrays.
Let NNZ denote the number of nonzero entries of M. The first array is A, which is of length
NNZ, and holds all nonzero entries of M in left-to-right top-to-bottom order. The second
array is IA, which is of length m + 1 (i.e., one entry per row, plus one). IA(i) contains
the index in A of the first nonzero element of row i. Row i of the original matrix extends
from A(IA(i)) to A(IA(i+1)-1). The third array, JA, contains the column index of each
element of A, so it also is of length NNZ.
For example, the matrix
[ 1 2 0 0 ]
[ 0 3 9 0 ]
[ 0 1 4 0 ]
is a three-by-four matrix with six nonzero elements, so
A = [ 1 2 3 9 1 4 ]
IA = [ 1 3 5 7 ]
JA = [ 1 2 2 3 2 3 ]
Another possibility is to use quadtrees.
Example
A bitmap image having only 2 colors, with one of them dominant (say a file that stores a
handwritten signature) can be encoded as a sparse matrix that contains only row and
column numbers for pixels with the non-dominant color. Diagonal matrices
A very efficient structure for a diagonal matrix is to store just the entries in the main
diagonal as a one-dimensional array, so a diagonal n×n matrix requires only n entries
Bandwidth
The lower bandwidth of a matrix A is the smallest number p such that the entry aij
vanishes whenever i > j + p. Similarly, the upper bandwidth is the smallest p such that aij
= 0 whenever i < j − p (Golub & Van Loan 1996, §1.2.1). For example, a tridiagonal
matrix has lower bandwidth 1 and upper bandwidth 1.
Matrices with small upper and lower bandwidth are known as band matrices and often
lend themselves to simpler algorithms than general sparse matrices; one can sometimes
apply dense matrix algorithms and simply loop over a reduced number of indices.
Reducing bandwidth
The Cuthill-McKee algorithm can be used to reduce the bandwidth of a sparse symmetric
matrix. There are, however, matrices for which the Reverse Cuthill-McKee algorithm
performs better.
The U.S. National Geodetic Survey (NGS) uses Dr. Richard Snay's "Banker's" algorithm
because on realistic sparse matrices used in Geodesy work it has better performance.
There are many other methods in use.
Reducing fill-in
"Fill-in" redirects here. For the puzzle, see Fill-In (puzzle).
The fill-in of a matrix are those entries which change from an initial zero to a non-zero
value during the execution of an algorithm. To reduce the memory requirements and the
number of arithmetic operations used during an algorithm it is useful to minimize the fillin by switching rows and columns in the matrix. The symbolic Cholesky decomposition
can be used to calculate the worst possible fill-in before doing the actual Cholesky
decomposition.
There are other methods than the Cholesky decomposition in use. Orthogonalization
methods (such as QR factorization) are common, for example, when solving problems by
least squares methods. While the theoretical fill-in is still the same, in practical terms the
"false non-zeros" can be different for different methods. And symbolic versions of those
algorithms can be used in the same manner as the symbolic Cholesky to compute worst
case fill-in.
Solving sparse matrix equations
Both iterative and direct methods exist for sparse matrix solving. One popular iterative
method is the conjugate gradient method.
Book