Slatec(3) User Contributed Perl Documentation Slatec(3)NAMEPDL::Slatec - PDL interface to the slatec numerical programming library
SYNOPSIS
use PDL::Slatec;
($ndeg, $r, $ierr, $a) = polyfit($x, $y, $w, $maxdeg, $eps);
DESCRIPTION
This module serves the dual purpose of providing an interface to parts
of the slatec library and showing how to interface PDL to an external
library. Using this library requires a fortran compiler; the source
for the routines is provided for convenience.
Currently available are routines to: manipulate matrices; calculate
FFT's; fit data using polynomials; and interpolate/integrate data using
piecewise cubic Hermite interpolation.
Piecewise cubic Hermite interpolation (PCHIP)
PCHIP is the slatec package of routines to perform piecewise cubic
Hermite interpolation of data. It features software to produce a
monotone and "visually pleasing" interpolant to monotone data.
According to Fritsch & Carlson ("Monotone piecewise cubic
interpolation", SIAM Journal on Numerical Analysis 17, 2 (April 1980),
pp. 238-246), such an interpolant may be more reasonable than a cubic
spline if the data contains both "steep" and "flat" sections.
Interpolation of cumulative probability distribution functions is
another application. These routines are cryptically named (blame
FORTRAN), beginning with 'ch', and accept either float or double
piddles.
Most of the routines require an integer parameter called "check"; if
set to 0, then no checks on the validity of the input data are made,
otherwise these checks are made. The value of "check" can be set to 0
if a routine such as chim has already been successfully called.
· If not known, estimate derivative values for the points using the
chim, chic, or chsp routines (the following routines require both
the function ("f") and derivative ("d") values at a set of points
("x")).
· Evaluate, integrate, or differentiate the resulting PCH function
using the routines: chfd; chfe; chia; chid.
· If desired, you can check the monotonicity of your data using chcm.
FUNCTIONS
eigsys
Eigenvalues and eigenvectors of a real positive definite symmetric
matrix.
($eigvals,$eigvecs) = eigsys($mat)
Note: this function should be extended to calculate only eigenvalues if
called in scalar context!
matinv
Inverse of a square matrix
($inv) = matinv($mat)
polyfit
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. Handles threading correctly--one can pass in a 2D PDL (as
$y) and it will pass back a 2D PDL, the rows of which are the
polynomial regression results (in $r corresponding to the rows of $y.
($ndeg, $r, $ierr, $a, $coeffs, $rms) = polyfit($x, $y, $w, $maxdeg, [$eps]);
$coeffs = polyfit($x,$y,$w,$maxdeg,[$eps]);
where on input:
$x and $y are the values to fit to a polynomial. $w are weighting
factors $maxdeg is the maximum degree of polynomial to use and $eps is
the required degree of fit.
and the output switches on list/scalar context.
In list context:
$ndeg is the degree of polynomial actually used $r is the values of the
fitted polynomial $ierr is a return status code, and $a is some working
array or other (preserved for historical purposes) $coeffs is the
polynomial coefficients of the best fit polynomial. $rms is the rms
error of the fit.
In scalar context, only $coeffs is returned.
Historically, $eps was modified in-place to be a return value of the
rms error. This usage is deprecated, and $eps is an optional parameter
now. It is still modified if present.
$a is a working array accessible to Slatec - you can feed it to several
other Slatec routines to get nice things out. It does not thread
correctly and should probably be fixed by someone. If you are reading
this, that someone might be you.
This version of polyfit handles bad values correctly. Bad values in $y
are ignored for the fit and give computed values on the fitted curve in
the return. Bad values in $x or $w are ignored for the fit and result
in bad elements in the output.
polycoef
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, $a);
polyvalue
Convenience wrapper routine around the "pvalue" "slatec" function.
Separates supplied arguments and return values.
For multiple input x positions, a corresponding y position is
calculated.
The derivatives PDL is one dimensional (of size "nder") if a single x
position is supplied, two dimensional if more than one x position is
supplied.
Use the coefficients generated by "polyfit" (or "polfit") to evaluate
the polynomial fit of degree "l", along with the first "nder" of its
derivatives, at a specified point.
($yfit, $yp) = polyvalue($l, $nder, $x, $a);
detslatec
compute the determinant of an invertible matrix
$mat = zeroes(5,5); $mat->diagonal(0,1) .= 1; # unity matrix
$det = detslatec $mat;
Usage:
$determinant = detslatec $matrix;
Signature: detslatec(mat(n,m); [o] det())
"detslatec" computes the determinant of an invertible matrix and barfs
if the matrix argument provided is non-invertible. The matrix threads
as usual.
This routine was previously known as "det" which clashes now with det
which is provided by PDL::MatrixOps. For the moment PDL::Slatec will
also load PDL::MatrixOps thereby making sure that older scripts work.
PDL::Slatec::fft
Fast Fourier Transform
$v_in = pdl(1,0,1,0);
($azero,$a,$b) = PDL::Slatec::fft($v_in);
"PDL::Slatec::fft" is a convenience wrapper for ezfftf, and performs a
Fast Fourier Transform on an input vector $v_in. The return values are
the same as for ezfftf.
PDL::Slatec::rfft
reverse Fast Fourier Transform
$v_out = PDL::Slatec::rfft($azero,$a,$b);
print $v_in, $vout
[1 0 1 0] [1 0 1 0]
"PDL::Slatec::rfft" is a convenience wrapper for ezfftb, and performs a
reverse Fast Fourier Transform. The input is the same as the output of
PDL::Slatec::fft, and the output of "rfft" is a data vector, similar to
what is input into PDL::Slatec::fft.
svdc
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
svdc does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
poco
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.
poco does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
geco
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.
geco does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
gefa
Signature: (a(n,n);int [o]ipvt(n);int [o]info())
Factor a matrix using Gaussian elimination.
gefa does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
podi
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.
podi does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
gedi
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.
gedi does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
gesl
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.
gesl does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
rs
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.
rs does not process bad values. It will set the bad-value flag of all
output piddles if the flag is set for any of the input piddles.
ezffti
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()".
ezffti does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
ezfftf
Signature: (r(n);[o]azero();[o]a(n);[o]b(n);wsave(foo))
ezfftf does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
ezfftb
Signature: ([o]r(n);azero();a(n);b(n);wsave(foo))
ezfftb does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
pcoef
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.
pcoef does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
pvalue
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.
pvalue does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chim
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 $x is not strictly increasing.
chim does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chic
Signature: (int ic(two=2);vc(two=2);mflag();x(n);f(n);[o]d(n);wk(nwk);int [o]ierr())
Calculate the derivatives of (x,f(x)) using cubic Hermite
interpolation.
Calculate the derivatives at the given points ("$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 $x is not strictly increasing.
· -4 if "abs(ic(0)) > 5".
· -5 if "abs(ic(1)) > 5".
· -6 if both -4 and -5 are true.
· -7 if "nwk < 2*(n-1)".
chic does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chsp
Signature: (int ic(two=2);vc(two=2);x(n);f(n);[o]d(n);wk(nwk);int [o]ierr())
Calculate the derivatives of (x,f(x)) using cubic spline interpolation.
Calculate the derivatives, using cubic spline interpolation, at the
given points ("$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 $x is not strictly increasing.
· -4 if "ic(0) < 0" or "ic(0) > 4".
· -5 if "ic(1) < 0" or "ic(1) > 4".
· -6 if both of the above are true.
· -7 if "nwk < 2*n".
· -8 in case of trouble solving the linear system for the interior
derivative values.
chsp does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chfd
Signature: (x(n);f(n);d(n);int check();xe(ne);[o]fe(ne);[o]de(ne);int [o]ierr())
Interpolate function and derivative values.
Given a piecewise cubic Hermite function - such as from chim - evaluate
the function ($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.
chfd does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chfe
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".
chfe does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chia
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.
· 1 if $a lies outside $x.
· 2 if $b lies outside $x.
· 3 if both 1 and 2 are true.
· -1 if "nelem($x) < 2"
· -3 if $x is not strictly increasing.
· -4 if an error has occurred in a lower-level routine, which should
never happen.
chia does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chid
Signature: (x(n);f(n);d(n);int check();int ia();int ib();[o]ans();int [o]ierr())
Integrate (x,f(x)) between data points.
Evaluate the definite integral of a a piecewise cubic Hermite function
between "x($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.
chid does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chcm
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.
chcm does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
chbs
Signature: (x(n);f(n);d(n);int knotyp();int nknots();t(tsize);[o]bcoef(bsize);int [o]ndim();int [o]kord();int [o]ierr())
Piecewise Cubic Hermite function to B-Spline converter.
The resulting B-spline representation of the data (i.e. "nknots", "t",
"bcoeff", "ndim", and "kord") can be evaluated by "bvalu" (which is
currently not available).
Array sizes: "tsize = 2*n + 4", "bsize = 2*n", and "ndim = 2*n".
"knotyp" is a flag which controls the knot sequence. The knot sequence
"t" is normally computed from $x by putting a double knot at each "x"
and setting the end knot pairs according to the value of "knotyp"
(where "m = ndim = 2*n"):
· 0 - Quadruple knots at the first and last points.
· 1 - Replicate lengths of extreme subintervals: "t( 0 ) = t( 1 ) =
x(0) - (x(1)-x(0))" and "t(m+3) = t(m+2) = x(n-1) +
(x(n-1)-x(n-2))"
· 2 - Periodic placement of boundary knots: "t( 0 ) = t( 1 ) = x(0)
- (x(n-1)-x(n-2))" and "t(m+3) = t(m+2) = x(n) + (x(1)-x(0))"
· <0 - Assume the "nknots" and "t" were set in a previous call.
"nknots" is the number of knots and may be changed by the routine. If
"knotyp >= 0", "nknots" will be set to "ndim+4", otherwise it is an
input variable, and an error will occur if its value is not equal to
"ndim+4".
"t" is the array of "2*n+4" knots for the B-representation and may be
changed by the routine. If "knotyp >= 0", "t" will be changed so that
the interior double knots are equal to the x-values and the boundary
knots set as indicated above, otherwise it is assumed that "t" was set
by a previous call (no check is made to verify that the data forms a
legitimate knot sequence).
Error status returned by $ierr:
· 0 if successful.
· -4 if "knotyp > 2".
· -5 if "knotyp < 0" and "nknots != 2*n + 4".
chbs does not process bad values. It will set the bad-value flag of
all output piddles if the flag is set for any of the input piddles.
polfit
Signature: (x(n); y(n); w(n); int maxdeg(); int [o]ndeg(); [o]eps(); [o]r(n); int [o]ierr(); [o]a(foo); [o]coeffs(bar);[t]xtmp(n);[t]ytmp(n);[t]wtmp(n);[t]rtmp(n))
Fit discrete data in a least squares sense by polynomials
in one variable. "x()", "y()" and "w()" must be of the same
type. This version handles bad values appropriately
polfit processes bad values. It will set the bad-value flag of all
output piddles if the flag is set for any of the input piddles.
AUTHOR
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.
perl v5.18.1 2014-01-17 Slatec(3)
