Curve Fitting Toolbox™ spline functions contain versions of the essential MATLAB® programs of the B-spline package (extended to handle also vector-valued splines) as described in A Practical Guide to Splines, (Applied Math. Sciences Vol. 27, Springer Verlag, New York (1978), xxiv + 392p; revised edition (2001), xviii+346p), hereafter referred to as PGS. The toolbox makes it easy to create and work with piecewise-polynomial functions.
The typical use envisioned for this toolbox involves the construction and subsequent use of a piecewise-polynomial approximation. This construction would involve data fitting, but there is a wide range of possible data that could be fit. In the simplest situation, one is given points (ti,yi) and is looking for a piecewise-polynomial function f that satisfies f(ti) = yi, all i, more or less. An exact fit would involve interpolation, an approximate fit might involve least-squares approximation or the smoothing spline. But the function to be approximated may also be described in more implicit ways, for example as the solution of a differential or integral equation. In such a case, the data would be of the form (Af)(ti), with A some differential or integral operator. On the other hand, one might want to construct a spline curve whose exact location is less important than is its overall shape. Finally, in all of this, one might be looking for functions of more than one variable, such as tensor product splines.
Care has been taken to make this work as painless and intuitive
as possible. In particular, the user need not worry about just how
splines are constructed or stored for later use, nor need the casual
user worry about such items as “breaks” or “knots”
or “coefficients”. It is enough to know that each function
constructed is just another variable that is freely usable as input
(where appropriate) to many of the commands, including all commands
fn, which stands for
At times, it may be also useful to know that, internal to the toolbox,
splines are stored in different forms, with the command
to convert between forms.
At present, the toolbox supports two major forms for the representation of piecewise-polynomial functions, because each has been found to be superior to the other in certain common situations. The B-form is particularly useful during the construction of a spline, while the ppform is more efficient when the piecewise-polynomial function is to be evaluated extensively. These two forms are almost exactly the B-representation and the pp representation used in A Practical Guide to Splines.
But, over the years, the Curve Fitting Toolbox spline functions have gone beyond the programs in A Practical Guide to Splines. The toolbox now supports the `scattered translates' form, or stform, in order to handle the construction and use of bivariate thin-plate splines, and also two ways to represent rational splines, the rBform and the rpform, in order to handle NURBS.
Splines can be very effective for data fitting because the linear
systems to be solved for this are banded, hence the work needed for
their solution, done properly, grows only linearly with the number
of data points. In particular, the MATLAB sparse matrix facilities
are used in the Curve Fitting Toolbox spline functions when that
is more efficient than the toolbox's own equation solver,
which relies on the fact that some of the linear systems here are
even almost block diagonal.
spline construction commands are equipped to produce bivariate (or
even multivariate) piecewise-polynomial functions as tensor products
of the univariate functions used here, and the various
also work for these multivariate functions.
There are various examples, all accessible through the Help browser. You are strongly urged to
have a look at some of them, or at the
splinetool, to help you
work with splines.
Consider the set
of all (scalar-valued) piecewise-polynomials of order k with breaks ξ1 < ... < ξl + 1 that, for i = 2...l, may have a jump across ξi in its μith derivative but have no jump there in any lower order derivative. This set is a linear space, in the sense that any scalar multiple of a function in S is again in S, as is the sum of any two functions in S.
In particular, according to the Curry-Schoenberg Theorem, our space S has a basis consisting of B-splines, namely the sequence of all B-splines of the form , j = 1...n, with the knot sequence t obtained from the break sequence ξ and the sequence µ by the following conditions:
Have both ξ1 and ξl + 1 occur in t exactly k times
For each i = 2:l, have ξi occur in t exactly k – µi times
Make sure the sequence is nondecreasing and only contains elements from ξ
Note the correspondence between the multiplicity of a knot and the smoothness of the spline across that knot. In particular, at a simple knot, that is a knot that appears exactly once in the knot sequence, only the (k – 1)st derivative may be discontinuous.
The MATLAB technical computing environment provides spline approximation via the
spline. If called in the form
returns the ppform of the cubic spline with break sequence
takes the value
i, and satisfies the not-a-knot end condition. In other
words, the command
cs = spline(x,y) gives the same
result as the command
available in the Curve Fitting Toolbox spline functions. But only the latter also works when
y describe multivariate gridded data. In
MATLAB, cubic spline interpolation to multivariate gridded data is provided
by the command
returns values of the interpolating tensor product cubic spline at the grid
Further, any of the Curve Fitting Toolbox spline
fn... commands can be applied to the output
of the MATLAB
spline(x,y) command, with simple versions of the Curve Fitting Toolbox spline commands
fnbrk available directly in MATLAB, as the commands
The Curve Fitting Toolbox spline functions started out as an extension of the MATLAB environment of interest to experts in spline approximation, to aid them in the construction and testing of new methods of spline approximation. Such people will have mastered the material in A Practical Guide to Splines.
However, the basic commands for constructing and using spline approximations are set up to be usable with no more knowledge than it takes to understand what it means to, say, construct an interpolant or a least squares approximant to some data, or what it means to differentiate or integrate a function.
With that in mind, there are sections, like Cubic Spline Interpolation, that are meant even for the novice, while sections devoted to a detailed example, like the one on constructing a Chebyshev spline or on constructing and using tensor products, are meant for users interested in developing their own spline commands.
The Curve Fitting Toolbox spline functions can handle vector-valued splines, i.e., splines whose values lie in Rd. Since MATLAB started out with just one variable type, that of a matrix, there is even now some uncertainty about how to deal with vectors, i.e., lists of numbers. MATLAB sometimes stores such a list in a matrix with just one row, and other times in a matrix with just one column. In the first instance, such a 1-row matrix is called a row-vector; in the second instance, such a 1-column matrix is called a column-vector. Either way, these are merely different ways for storing vectors, not different kinds of vectors.
A point in Rd, i.e., a d-vector, is always stored
as a column vector. In particular, if you want to supply an
d-vectors to one of the commands, you are expected
to provide that list as the
n columns of a matrix
While other lists of numbers (e.g., a knot sequence or a break sequence) may be stored internally as row vectors, you may supply such lists as you please, as a row vector or a column vector.
Most of the spline commands in this toolbox have names that follow one of the following patterns:
cs... commands construct cubic splines (in
sp... commands construct splines in B-form
fn... commands operate on spline functions
..2... commands convert something
..api commands construct an approximation
..aps commands construct an approximation
..ap2 commands construct a least-squares
...knt commands construct (part of) a particular
...dem commands are examples.
For ease of use, most Curve Fitting Toolbox spline functions
have default arguments. In the reference entry under Syntax, we usually
first list the function with all necessary input
arguments and then with all possible input arguments.
When there is more than one optional argument, then, sometimes, but
not always, their exact order is immaterial. When their order does
matter, you have to specify every optional argument preceding the
one(s) you are interested in. In this situation, you can specify the
default value for an optional argument by using
empty matrix) as the input for it. The description in the reference
page tells you the default value for each optional input argument.
As in MATLAB, only the output arguments explicitly specified are returned to the user.
MathWorks® would like to acknowledge the contributions of Carl de Boor to the Curve Fitting Toolbox spline functions. Professor de Boor authored the Spline Toolbox™ from its first release until Version 3.3.4 (2008).
Professor de Boor received the John von Neumann Prize in 1996 and the National Medal of Science in 2003. He is a member of both the American Academy of Arts and Sciences and the National Academy of Sciences. He is the author of A Practical Guide to Splines (Springer, 2001).
Some of the spline function naming conventions are the result of a discussion with Jörg Peters, then a graduate student in Computer Sciences at the University of Wisconsin-Madison.