PDL::Slatec - PDL interface to the slatec numerical programming library
use PDL::Slatec;
($ndeg, $r, $ierr, $a) = polyfit($x, $y, $w, $maxdeg, $eps);
This module serves the dual purpose of providing an interface to parts of the slatec library and showing how to interface PDL to an external library. Using this library requires a fortran compiler; the source for the routines is provided for convenience.
Currently available are routines to: manipulate matrices; calculate FFT's; fit data using polynomials; and interpolate/integrate data using piecewise cubic Hermite interpolation.
PCHIP is the slatec package of routines to perform piecewise cubic Hermite interpolation of data. It features software to produce a monotone and ``visually pleasing'' interpolant to monotone data. According to Fritsch & Carlson (``Monotone piecewise cubic interpolation'', SIAM Journal on Numerical Analysis 17, 2 (April 1980), pp. 238-246), such an interpolant may be more reasonable than a cubic spline if the data contains both ``steep'' and ``flat'' sections. Interpolation of cumulative probability distribution functions is another application. These routines are cryptically named (blame FORTRAN), beginning with 'ch', and accept either float or double piddles.
Most of the routines require an integer parameter called check
; if set to 0, then no checks on the validity of the input data are made,
otherwise these checks are made. The value of check
can be set to 0 if a routine such as chim has already been successfully called.
If not known, estimate derivative values for the points using the chim, chic, or chsp routines (the following routines require both the function (f
) and derivative (d
) values at a set of points (x)).
Evaluate, integrate, or differentiate the resulting PCH function using the routines: chfd; chfe; chia; chid.
If desired, you can check the monotonicity of your data using chcm.
Eigenvalues and eigenvectors of a real positive definite symmetric matrix.
($eigvals,$eigvecs) = eigsys($mat)
Note: this function should be extended to calculate only eigenvalues if called in scalar context!
Inverse of a square matrix
($inv) = matinv($mat)
Convenience wrapper routine about the polfit
slatec
function. Separates supplied arguments and return values.
Fit discrete data in a least squares sense by polynomials in one variable.
($ndeg, $r, $ierr, $a) = polyfit($x, $y, $w, $maxdeg, $eps);
eps
is modified to contain the rms error of the fit.
Convenience wrapper routine around the pcoef
slatec
function. Separates supplied arguments and return values.
Convert the polyfit
/polfit
coefficients to Taylor series form.
$tc = polycoef($l, $c, $a);
Convenience wrapper routine around the pvalue
slatec
function. Separates supplied arguments and return values.
For multiple input x positions, a corresponding y position is calculated.
The derivatives PDL is one dimensional (of size nder
) if a single x position is supplied, two dimensional if more than one x
position is supplied.
Use the coefficients generated by polyfit
(or polfit
) to evaluate the polynomial fit of degree l
, along with the first nder
of its derivatives, at a specified point.
($yfit, $yp) = polyvalue($l, $nder, $x, $a);
Signature: (x(n,p);[o]s(p);[o]e(p);[o]u(n,p);[o]v(p,p);[o]work(n);int job();int [o]info())
singular value decomposition of a matrix
Signature: (a(n,n);rcond();[o]z(n);int [o]info())
Factor a real symmetric positive definite matrix and estimate the condition number of the matrix.
Signature: (a(n,n);int [o]ipvt(n);[o]rcond();[o]z(n))
Factor a matrix using Gaussian elimination and estimate the condition number of the matrix.
Signature: (a(n,n);int [o]ipvt(n);int [o]info())
Factor a matrix using Gaussian elimination.
Signature: (a(n,n);[o]det(two=2);int job())
Compute the determinant and inverse of a certain real symmetric positive definite matrix using the factors computed by poco.
Signature: (a(n,n);int [o]ipvt(n);[o]det(two=2);[o]work(n);int job())
Compute the determinant and inverse of a matrix using the factors computed by geco or gefa.
Signature: (a(lda,n);int ipvt(n);b(n);int job())
Solve the real system A*X=B
or TRANS(A)*X=B
using the factors computed by geco or gefa.
Signature: (a(n,n);[o]w(n);int matz();[o]z(n,n);[t]fvone(n);[t]fvtwo(n);int [o]ierr())
This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) of a REAL SYMMETRIC matrix.
Signature: (int n();[o]wsave(foo))
Subroutine ezffti initializes the work array wsave()
which is used in both ezfftf and
ezfftb. The prime factorization of n
together with a tabulation of the trigonometric functions are computed and
stored in wsave()
.
Signature: (r(n);[o]azero();[o]a(n);[o]b(n);wsave(foo))
Signature: ([o]r(n);azero();a(n);b(n);wsave(foo))
Signature: (x(n);y(n);w(n);int maxdeg();int [o]ndeg();[o]eps();[o]r(n);int [o]ierr();[o]a(foo))
Fit discrete data in a least squares sense by polynomials in one variable. x(), y()
and w()
must be of the same type.
Signature: (int l();c();[o]tc(bar);a(foo))
Convert the polfit
coefficients to Taylor series form.
c
and a() must be of the same type.
Signature: (int l();x();[o]yfit();[o]yp(nder);a(foo))
Use the coefficients generated by polfit
to evaluate the polynomial fit of degree l
, along with the first nder
of its derivatives, at a specified point. x and a must be of the same type.
Signature: (x(n);f(n);[o]d(n);int [o]ierr())
Calculate the derivatives of (x,f(x)) using cubic Hermite interpolation.
Calculate the derivatives at the given set of points ($x,$f
, where $x is strictly increasing). The resulting set of points - $x,$f,$d
, referred to as the cubic Hermite representation - can then be used in
other functions, such as chfe, chfd, and chia.
The boundary conditions are compatible with monotonicity, and if the data are only piecewise monotonic, the interpolant will have an extremum at the switch points; for more control over these issues use chic.
Error status returned by $ierr
:
0 if successful.
> 0 if there were ierr
switches in the direction of monotonicity (data still valid).
-1 if nelem($x) < 2
.
-3 if $x is not strictly increasing.
Signature: (int ic(two=2);vc(two=2);mflag();x(n);f(n);[o]d(n);wk(nwk);int [o]ierr())
Calculate the derivatives of (x,f(x)) using cubic Hermite interpolation.
Calculate the derivatives at the given points ($x,$f
, where $x is strictly increasing). Control over the boundary conditions is given by
the
$ic
and $vc
piddles, and the value of $mflag
determines the treatment of points where monotoncity switches direction. A
simpler, more restricted, interface is available using chim.
The first and second elements of $ic
determine the boundary conditions at the start and end of the data
respectively. If the value is 0, then the default condition, as used by
chim, is adopted. If greater than zero, no adjustment for monotonicity is made,
otherwise if less than zero the derivative will be adjusted. The allowed
magnitudes for ic(0)
are:
1 if first derivative at x(0) is given in vc(0)
.
2 if second derivative at x(0) is given in vc(0)
.
3 to use the 3-point difference formula for d(0)
. (Reverts to the default b.c. if n < 3
)
4 to use the 4-point difference formula for d(0)
. (Reverts to the default b.c. if n < 4
)
5 to set d(0)
so that the second derivative is continuous at x(1). (Reverts to the default b.c. if n < 4
)
The values for ic(1)
are the same as above, except that the first-derivative value is stored in vc(1)
for cases 1 and 2. The values of $vc
need only be set if options 1 or 2 are chosen for $ic
.
Set $mflag = 0
if interpolant is required to be monotonic in each interval, regardless of
the data. This causes $d
to be set to 0 at all switch points. Set $mflag
to be non-zero to use a formula based on the 3-point difference formula at
switch points. If $mflag > 0
, then the interpolant at swich points is forced to not deviate from the
data by more than $mflag*dfloc
, where dfloc
is the maximum of the change of $f
on this interval and its two immediate neighbours. If $mflag < 0
, no such control is to be imposed.
The piddle $wk
is only needed for work space. However, I could not get it to work as a
temporary variable, so you must supply it; it is a 1D piddle with 2*n
elements.
Error status returned by $ierr
:
0 if successful.
1 if ic(0) < 0
and d(0)
had to be adjusted for monotonicity.
2 if ic(1) < 0
and d(n-1)
had to be adjusted for monotonicity.
3 if both 1 and 2 are true.
-1 if n < 2
.
-3 if $x is not strictly increasing.
-4 if abs(ic(0)) > 5
.
-5 if abs(ic(1)) > 5
.
-6 if both -4 and -5 are true.
-7 if nwk < 2*(n-1)
.
Signature: (int ic(two=2);vc(two=2);x(n);f(n);[o]d(n);wk(nwk);int [o]ierr())
Calculate the derivatives of (x,f(x)) using cubic spline interpolation.
Calculate the derivatives, using cubic spline interpolation, at the given
points ($x,$f
), with the specified boundary conditions. Control over the boundary
conditions is given by the
$ic
and $vc
piddles. The resulting values - $x,$f,$d
- can be used in all the functions which expect a cubic Hermite function.
The first and second elements of $ic
determine the boundary conditions at the start and end of the data
respectively. The allowed values for ic(0)
are:
0 to set d(0)
so that the third derivative is continuous at x(1).
1 if first derivative at x(0) is given in vc(0
).
2 if second derivative at x(0
) is given in vc(0)
.
3 to use the 3-point difference formula for d(0)
. (Reverts to the default b.c. if n < 3
.)
4 to use the 4-point difference formula for d(0)
. (Reverts to the default b.c. if n < 4
.)
The values for ic(1)
are the same as above, except that the first-derivative value is stored in vc(1)
for cases 1 and 2. The values of $vc
need only be set if options 1 or 2 are chosen for $ic
.
The piddle $wk
is only needed for work space. However, I could not get it to work as a
temporary variable, so you must supply it; it is a 1D piddle with 2*n
elements.
Error status returned by $ierr
:
0 if successful.
-1 if nelem($x) < 2
.
-3 if $x is not strictly increasing.
-4 if ic(0) < 0
or ic(0) > 4
.
-5 if ic(1) < 0
or ic(1) > 4
.
-6 if both of the above are true.
-7 if nwk < 2*n
.
-8 in case of trouble solving the linear system for the interior derivative values.
Signature: (x(n);f(n);d(n);int check();xe(ne);[o]fe(ne);[o]de(ne);int [o]ierr())
Interpolate function and derivative values.
Given a piecewise cubic Hermite function - such as from
chim - evaluate the function ($fe
) and derivative ($de
) at a set of points ($xe
). If function values alone are required, use chfe. Set check
to 0 to skip checks on the input data.
Error status returned by $ierr
:
0 if successful.
>0 if extrapolation was performed at ierr
points (data still valid).
-1 if nelem($x) < 2
-3 if $x is not strictly increasing.
-4 if nelem($xe) < 1
.
-5 if an error has occurred in a lower-level routine, which should never happen.
Signature: (x(n);f(n);d(n);int check();xe(ne);[o]fe(ne);int [o]ierr())
Interpolate function values.
Given a piecewise cubic Hermite function - such as from
chim - evaluate the function ($fe
) at a set of points ($xe
). If derivative values are also required, use chfd. Set check
to 0 to skip checks on the input data.
Error status returned by $ierr
:
0 if successful.
>0 if extrapolation was performed at ierr
points (data still valid).
-1 if nelem($x) < 2
-3 if $x is not strictly increasing.
-4 if nelem($xe) < 1
.
Signature: (x(n);f(n);d(n);int check();a();b();[o]ans();int [o]ierr())
Integrate (x,f(x)) over arbitrary limits.
Evaluate the definite integral of a a piecewise cubic Hermite function over
an arbitrary interval, given by [$a,$b]
. See chid if the integration limits are data points. Set check
to 0 to skip checks on the input data.
The values of $a and $b
do not have to lie within $x, although the resulting integral value will be highly suspect if they are
not.
Error status returned by $ierr
:
0 if successful.
2 if $b
lies outside $x.
3 if both 1 and 2 are true.
-1 if nelem($x) < 2
-3 if $x is not strictly increasing.
-4 if an error has occurred in a lower-level routine, which should never happen.
Signature: (x(n);f(n);d(n);int check();int ia();int ib();[o]ans();int [o]ierr())
Integrate (x,f(x)) between data points.
Evaluate the definite integral of a a piecewise cubic Hermite function between x($ia) and x($ib).
See chia for integration between arbitrary limits.
Although using a fortran routine, the values of
$ia
and $ib
are zero offset. Set check
to 0 to skip checks on the input data.
Error status returned by $ierr
:
0 if successful.
-1 if nelem($x) < 2
.
-3 if $x is not strictly increasing.
-4 if $ia
or $ib
is out of range.
Signature: (x(n);f(n);d(n);int check();int [o]ismon(n);int [o]ierr())
Check the given piecewise cubic Hermite function for monotonicity.
The outout piddle $ismon
indicates over which intervals the function is monotonic. Set check
to 0 to skip checks on the input data.
For the data interval [x(i),x(i+1)]
, the values of ismon(i)
can be:
-3 if function is probably decreasing
-1 if function is strictly decreasing
0 if function is constant
1 if function is strictly increasing
2 if function is non-monotonic
3 if function is probably increasing
If abs(ismon(i)) == 3
, the derivative values are near the boundary of the monotonicity region. A
small increase produces non-monotonicity, whereas a decrease produces
strict monotonicity.
The above applies to i = 0 .. nelem($x)-1
. The last element of
$ismon
indicates whether the entire function is monotonic over $x.
Error status returned by $ierr
:
0 if successful.
-1 if n < 2
.
-3 if $x is not strictly increasing.
Copyright (C) 1997 Tuomas J. Lukka. Copyright (C) 2000 Tim Jenness, Doug Burke. All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation under certain conditions. For details, see the file COPYING in the PDL distribution. If this file is separated from the PDL distribution, the copyright notice should be included in the file.