Index of Section 1 Manual Pages
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SPLINE(1) GNU Plotting Utilities SPLINE(1)
NAME
spline - interpolate datasets using splines under tension
SYNOPSIS
spline [ options ] [ files ]
DESCRIPTION
spline reads datasets from standard input or from one or
more files, and fits a smooth curve (a "spline") through
each dataset. An interpolated version of each dataset,
consisting of points from the smooth curve, is written to
standard output.
Unless the -a or -A options are used (see below), each
dataset should be a sequence of values for a vector-valued
function of a single scalar variable. That is, each
dataset should be a sequence of data points, given as
alternating t and y values. t is a scalar independent
variable, and y is a vector-valued dependent variable.
The dimensionality of y is specified with the -d option
(the default dimensionality is 1). Between each data
point and the next, t should increase.
An input file may contain more than a single dataset. If
an input file is in ASCII format (the default), its
datasets should be separated by blank lines. The t and y
values of the data points in each dataset may be arranged
arbitrarily, so long as they are separated by white space.
Besides datasets, an input file may contain any number of
comment lines, which should begin with the comment charac-
ter `#'. Comment lines are ignored. They are not treated
as blank, i.e., they do not interrupt a dataset in
progress.
Options and file names may be interspersed on the command
line, but the options are processed before the file names
are read. If -- is seen, it is interpreted as the end of
the options. If no file names are specified, or the file
name - is encountered, the standard input is read.
The type of interpolation, and the format of the input and
output files, may be selected by command-line options.
OPTIONS
Interpolation-Related Options
-f
--filter
Use a local interpolation algorithm (the cubic
Bessel algorithm), so that spline can be used as a
real-time 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 the default (global)
interpolation algorithm will be used.
-k k
--boundary-condition k
Set the boundary condition parameter for each con-
structed spline to be k. (The default value is
1.0.) In each of its components, the spline will
satisfy the two boundary conditions y"[0]=ky"[1]
and y"[n]=ky"[n-1]. Here y[0] and y[1] signify the
values of a specified component of the vector-val-
ued dependent variable y at the first two points of
a dataset, and y[n-1] 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
--number-of-intervals n
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. The default value for n is 100.
-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
Each interpolating curve will be a spline under
tension. This option sets the tension value (the
default is 0.0).
If tension equals zero, the curve will be a piece-
wise cubic spline. Increasing the tension above
zero makes the curve "tighter", and reduces the
likelihood of spurious inflection points. That is
because between each pair of successive points in a
dataset, the curve will satisfy the fourth-order
differential equation y""=sgn(tension)*(ten-
sion^2)y" in each of its components. As tension
increases to positive infinity, it will converge to
a polygonal line. The -T option may not be used if
-f is specified.
-t tmin tmax [tspacing]
--t-spacing 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 subin-
tervals 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 filter-like behavior (see above).
Format-Related Options
-d dimension
--y-dimension dimension
Set the dimensionality of the dependent variable y
in the input and output files to be dimension. The
default dimension is 1.
-I data-format
--input-format data-format
Set the data format for the input file(s) to be
data-format, which may be one of the following.
a ASCII format (the default). Each file is a
sequence of floating point numbers, inter-
preted as the t and y coordinates of the
successive data points in a dataset. If y
is d-dimensional, 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 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 d-dimensional, there will be d+1 num-
bers for each point. Successive datasets
are separated by a single occurrence of the
quantity FLT_MAX, which is the largest pos-
sible single precision floating point num-
ber. On most machines this is approximately
3.4x10^38.
d 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 d-dimensional, there
will be d+1 numbers for each point. Succes-
sive datasets are separated by a single
occurrence of the quantity DBL_MAX, which is
the largest possible double precision float-
ing point number. On most machines this is
approximately 1.8x10^308.
i 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 d-dimensional,
there will be d+1 numbers for each point.
Successive datasets are separated by a sin-
gle occurrence of the quantity INT_MAX,
which is the largest possible integer. On
most machines this is 2^31-1.
-a [step_size [lower_limit]]
--auto-abscissa [step_size [lower_limit]]
Automatically generate values for t, the indepen-
dent variable (the default values of step_size and
lower_limit are 1.0 and 0.0, respectively).
Irrespective of data format (`a', `f', `d', or
`i'), this option specifies that the values of t
are missing from the input file: the dataset(s) to
be read contain only values of y, the dependent
variable. So if y is d-dimensional, 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. This option is
useful, e.g., when interpolating curves rather than
functions.
-A
--auto-dist-abscissa
Automatically generate values for t, the indepen-
dent variable. This is a variant form of the -a
option. The increment from each t value to the
next will be the distance in d-dimensional space
between the corresponding y values, and the first t
value will be 0.0. That is, t will be "polygonal
arclength". This option is useful when interpolat-
ing curves rather than functions.
-O data-format
--output-format data-format
Set the data format for the output file to be data-
format. The interpretation of data-format is the
same as for the -I option. The default is `a',
i.e., ASCII format.
-P significant-digits
--precision significant-digits
Set the numerical precision for the t and y values
in the output file to be significant-digits. This
takes effect only if the output file is written in
`a' format, i.e., in ASCII. significant-digits
must be a positive integer (the default is 6).
-s
--suppress-abscissa
Omit the independent variable t from the output
file; for each point, supply only the dependent
variable y. If y is d-dimensional, there will be
only d numbers for each point, not d+1. This
option is useful when interpolating curves rather
than functions.
Informational Options
--help Print a list of command-line options, and exit.
--version
Print the version number of spline and the plotting
utilities package, and exit.
EXAMPLES
Typing
echo 0 0 1 1 2 0 | spline
will produce on standard output an interpolated dataset
consisting of 101 data points. If graphed, this interpo-
lated dataset will yield a parabola.
It is sometimes useful to interpolate between a sequence
of arbitrarily placed points in d-dimensional space, i.e.,
to "spline a curve" rather than a function. The -a and -s
options are used for this. For example,
echo 0 0 1 0 1 1 0 1 | spline -d 2 -a -s
will produce on standard output a 101-point dataset that
interpolates between the four points (0,0), (1,0), (1,1),
and (0,1). The -d 2 option specifies that the dependent
variable y is two-dimensional. The -a option specifies
that the t values are missing from the input and should be
automatically generated. The -s option specifies that the
t values should be stripped from the output.
AUTHORS
spline was written by Robert S. Maier (rsm@math.ari-
zona.edu), starting with an earlier version by Rich Mur-
phey (rich@freebsd.org). The algorithms for constructing
splines under tension are similar to those used in the
FITPACK subroutine library, and are ultimately due to Alan
K. Cline (cline@cs.utexas.edu).
SEE ALSO
"The GNU Plotting Utilities Manual".
BUGS
Email bug reports to bug-gnu-utils@gnu.org.
FSF Dec 1998 SPLINE(1)