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The spline
program will interpolate vectorvalued functions of a
scalar variable t, and curves in ddimensional space.
The algorithms used by spline
are similar to those discussed in
D. Kincaid and [E.] W. Cheney, Numerical Analysis (2nd
ed., Brooks/Cole, 1996), section 6.4, and C. de Boor, A
Practical Guide to Splines (SpringerVerlag, 1978), Chapter 4.
Input file names may be specified anywhere on the command line. That
is, the relative order of font names and commandline options does not
matter. If no file names are specified, or the file name `'
is specified, the standard input is read.
An input file may contain more than a single dataset. Unless the
`a' or `A' options are used (see below), each dataset is
expected to consist of a sequence of data points, given as alternating
t and y values. t is the scalar
independent variable, and y is the vectorvalued dependent
variable. The dimensionality of y is specified with the
`d' option (the default is 1).
If the input file is in ASCII format (the default), its datasets are
separated by blank lines. An input file may also contain any number of
comment lines, which must begin with the comment character `#'.
Comment lines are ignored. They are not treated as blank, i.e., they do
not interrupt a dataset in progress.
The options to spline
are listed below. There are three sorts of
option:

Options specifying the type of interpolation to be performed on each dataset.

Options specifying the input or output format.

Options requesting information (e.g., `help').
Options that take an argument are followed, in parentheses, by the type
and default value of the argument.
The following options specify the type of interpolation to be performed
on each dataset.
 `f'

 `filter'

Use a local interpolation algorithm (the cubic Bessel algorithm), so
that
spline
can be used as a realtime filter. The slope of the
interpolating curve at each point in a dataset will be chosen by fitting
a quadratic function through that point and the two adjacent points in
the dataset. If `f' is specified then the `t' option,
otherwise optional, must be used as well. Also, if `f' is
specified then the `k', `p', and `T' options may not
be used.
If `f' is not specified, then a different (global)
interpolation algorithm will be used.
 `k k'

 `boundarycondition k'

(Float, default 1.0.) Set the boundary condition parameter for each
constructed spline to be k. In each of its components, the
spline will satisfy the two boundary conditions y"[0]=ky"[1]
and y"[n]=ky"[n1]. Here y[0] and y[1] signify
the values of a specified component of the vectorvalued dependent
variable y at the first two points of a dataset, and
y[n1] and y[n] the values at the last two points.
Setting k to zero will yield a `natural' spline, i.e., one that
has zero curvature at the two ends of the dataset. The `k' option
may not be used if `f' or `p' is specified.
 `n n'

 `numberofintervals n'

(Positive integer, default 100.) Subdivide the interval over which
interpolation occurs into n subintervals. The number of data
points computed, and written to the output, will be n+1.
 `p'

 `periodic'

Construct a periodic spline. If this option is specified, the y
values for the first and last points in each dataset must be equal. The
`f' and `k' options may not be used if `p' is
specified.
 `T tension'

 `tension tension'

(Float, default 0.0.) Set the tension in each interpolating spline to
be tension. Between each pair of successive points in a dataset,
the constructed spline will satisfy the differential equation
@ifnottex
y""=sgn(tension)*(tension^2)y"
in each of its components. If tension equals zero, the spline
will be piecewise cubic. As tension increases to positive
infinity, the spline will converge to a polygonal line. The `T'
option may not be used if `f' is specified.
 `t tmin tmax [tspacing]'

 `tlimits tmin tmax [tspacing]'

For each dataset, set the interval over which interpolation occurs to be
the interval between tmin and tmax. If tspacing
is not specified, the interval will be divided into the number of
subintervals specified by the `n' option. If the `t'
option is not used, the interval over which interpolation occurs will be
the entire range of the independent variable in the dataset. The
`t' option must always be used if the `f' option is used to
request filterlike behavior (see above).
The following options specify the format of the input file(s) and the
output file.
 `d dimension'

 `ydimension dimension'

(Integer, default 1.) Set the dimensionality of the dependent variable
y in the input and output files to be dimension.
 `I dataformat'

 `inputformat dataformat'

(Character, default `a'.) Set the data format for the input file(s)
to be dataformat. The possible data formats are as follows.
 `a'

ASCII format. Each file is a sequence of floating point numbers,
interpreted as the t and y coordinates of the
successive data points in a dataset. If y is
ddimensional, there will be d+1 numbers for each point.
The t and y coordinates of a point need not appear on
the same line, and points need not appear on different lines. But if a
blank line occurs (i.e., two newlines in succession are seen), it is
interpreted as the end of a dataset, and the beginning of the next.
 `f'

@ifnottex
Single precision binary format. Each file is a sequence of floating
point numbers, interpreted as the t and y coordinates
of the successive data points in a dataset. If y is
ddimensional, there will be d+1 numbers for each point.
Successive datasets are separated by a single occurrence of the quantity
FLT_MAX
, which is the largest possible single precision floating
point number. On most machines this is approximately 3.4x10^38.
 `d'

@ifnottex
Double precision binary format. Each file is a sequence of double
precision floating point numbers, interpreted as the t and
y coordinates of the successive data points in a dataset. If
y is ddimensional, there will be d+1 numbers for
each point. Successive datasets are separated by a single occurrence of
the quantity
DBL_MAX
, which is the largest possible double
precision floating point number. On most machines this is
approximately 1.8x10^308.
 `i'

@ifnottex
Integer binary format. Each file is a sequence of integers, interpreted
as the t and y coordinates of the successive data
points in a dataset. If y is ddimensional, there will be
d+1 numbers for each point. Successive datasets are separated by
a single occurrence of the quantity
INT_MAX
, which is the largest
possible integer. On most machines this is 2^311.
 `a [step_size [lower_limit]]'

 `autoabscissa [step_size [lower_limit]]'

(Floats, defaults 1.0 and 0.0.) Automatically generate values for the
independent variable (t). Irrespective of data format
(`a', `f', `d', or `i'), this option specifies
that the values of the independent variable (t) are missing from
the input file: the dataset(s) to be read contain only values of the
dependent variable (y), so that if y is
ddimensional, there will be only d numbers for each
point. The increment from each t value to the next will be
step_size, and the first t value will be
lower_limit.
 `A'

 `autodistabscissa'

Automatically generate values for the independent variable
(t). This is a variant form of the `a' option. The
increment from each t value to the next will be the distance
between the corresponding y values, and the first t
value will be 0.0. This option is useful when interpolating curves
rather than functions (see section Advanced use of
spline
).
 `O dataformat'

 `outputformat dataformat'

(Character, default `a'.) Set the data format for the output file
to be dataformat. The interpretation of the dataformat
argument is the same as for the `I' option.
 `P significantdigits'

 `precision significantdigits'

(Positive integer, default 6.) Set the numerical precision for the
t and y values in the output file to be
significantdigits. This takes effect only if the output file is
written in `a' format, i.e., in ASCII.
 `s'

 `suppressabscissa'

Omit the independent variable t from the output file; for each
point, supply only the dependent variable y. If y is
ddimensional, there will be only d numbers for each
point, not d+1. This option is useful when interpolating
curves rather than functions (see section Advanced use of
spline
).
The following options request information.
 `help'

Print a list of commandline options, and then exit.
 `version'

Print the version number of
spline
and the plotting utilities
package, and exit.
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