Provided by: pdl_2.081-1_amd64 
NAME
PDL::Slatec - PDL interface to the slatec numerical programming library
SYNOPSIS
use PDL::Slatec;
($ndeg, $r, $ierr, $c) = polyfit($x, $y, $w, $maxdeg, $eps);
DESCRIPTION
This module serves the dual purpose of providing an interface to parts of the slatec
library and showing how to interface PDL to an external library. Using this library
requires a fortran compiler; the source for the routines is provided for convenience.
Currently available are routines to: manipulate matrices; calculate FFT's; fit data using
polynomials; and interpolate/integrate data using piecewise cubic Hermite interpolation.
Piecewise cubic Hermite interpolation (PCHIP)
PCHIP is the slatec package of routines to perform piecewise cubic Hermite interpolation
of data. It features software to produce a monotone and "visually pleasing" interpolant
to monotone data. According to Fritsch & Carlson ("Monotone piecewise cubic
interpolation", SIAM Journal on Numerical Analysis 17, 2 (April 1980), pp. 238-246), such
an interpolant may be more reasonable than a cubic spline if the data contains both
"steep" and "flat" sections. Interpolation of cumulative probability distribution
functions is another application. These routines are cryptically named (blame FORTRAN),
beginning with 'ch', and accept either float or double ndarrays.
Most of the routines require an integer parameter called "check"; if set to 0, then no
checks on the validity of the input data are made, otherwise these checks are made. The
value of "check" can be set to 0 if a routine such as "chim" has already been successfully
called.
• If not known, estimate derivative values for the points using the "chim", "chic", or
"chsp" routines (the following routines require both the function ("f") and derivative
("d") values at a set of points ("x")).
• Evaluate, integrate, or differentiate the resulting PCH function using the routines:
"chfd"; "chfe"; "chia"; "chid".
• If desired, you can check the monotonicity of your data using "chcm".
FUNCTIONS
eigsys
Eigenvalues and eigenvectors of a real positive definite symmetric matrix.
($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
broadcasting 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, $c, $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 $c 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.
$c is a working array accessible to Slatec - you can feed it to several other Slatec
routines to get nice things out. It does not broadcast 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, $x);
polyvalue
Convenience wrapper routine around the "pvalue" "slatec" function. Separates supplied
arguments and return values.
For multiple input x positions, a corresponding y position is calculated.
The derivatives PDL is one dimensional (of size "nder") if a single x position is
supplied, two dimensional if more than one x position is supplied.
Use the coefficients "c" generated by "polyfit" (or "polfit") to evaluate the polynomial
fit of degree "l", along with the first "nder" of its derivatives, at a specified point
"x".
($yfit, $yp) = polyvalue($l, $nder, $x, $c);
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 broadcasts as usual.
This routine was previously known as "det" which clashes now with det which is provided by
PDL::MatrixOps.
fft
Fast Fourier Transform
$v_in = pdl(1,0,1,0);
($azero,$x,$y) = 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".
rfft
reverse Fast Fourier Transform
$v_out = PDL::Slatec::rfft($azero,$x,$y);
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);longlong job();longlong [o]info())
singular value decomposition of a matrix
svdc does not process bad values. It will set the bad-value flag of all output ndarrays
if the flag is set for any of the input ndarrays.
poco
Signature: (a(n,n);rcond();[o]z(n);longlong [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 ndarrays
if the flag is set for any of the input ndarrays.
geco
Signature: (a(n,n);longlong [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 ndarrays
if the flag is set for any of the input ndarrays.
gefa
Signature: (a(n,n);longlong [o]ipvt(n);longlong [o]info())
Factor a matrix using Gaussian elimination.
gefa does not process bad values. It will set the bad-value flag of all output ndarrays
if the flag is set for any of the input ndarrays.
podi
Signature: (a(n,n);[o]det(two=2);longlong 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 ndarrays
if the flag is set for any of the input ndarrays.
gedi
Signature: (a(n,n);longlong [o]ipvt(n);[o]det(two=2);[o]work(n);longlong 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 ndarrays
if the flag is set for any of the input ndarrays.
gesl
Signature: (a(lda,n);longlong ipvt(n);b(n);longlong 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 ndarrays
if the flag is set for any of the input ndarrays.
rs
Signature: (a(n,n);[o]w(n);longlong matz();[o]z(n,n);[t]fvone(n);[t]fvtwo(n);longlong [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 ndarrays if
the flag is set for any of the input ndarrays.
ezffti
Signature: (longlong 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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
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 ndarrays
if the flag is set for any of the input ndarrays.
pcoef
Signature: (longlong 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 ndarrays
if the flag is set for any of the input ndarrays.
pvalue
Signature: (longlong 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 ndarrays
if the flag is set for any of the input ndarrays.
chim
Signature: (x(n);f(n);[o]d(n);longlong [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 $x is not strictly increasing.
chim does not process bad values. It will set the bad-value flag of all output ndarrays
if the flag is set for any of the input ndarrays.
chic
Signature: (longlong ic(two=2);vc(two=2);mflag();x(n);f(n);[o]d(n);wk(nwk);longlong [o]ierr())
Calculate the derivatives of (x,f(x)) using cubic Hermite interpolation.
Calculate the derivatives at the given points ("$x,$f", where $x is strictly increasing).
Control over the boundary conditions is given by the $ic and $vc ndarrays, 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 ndarray $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 ndarray 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 $x is not strictly increasing.
• -4 if "abs(ic(0)) > 5".
• -5 if "abs(ic(1)) > 5".
• -6 if both -4 and -5 are true.
• -7 if "nwk < 2*(n-1)".
chic does not process bad values. It will set the bad-value flag of all output ndarrays
if the flag is set for any of the input ndarrays.
chsp
Signature: (longlong ic(two=2);vc(two=2);x(n);f(n);[o]d(n);wk(nwk);longlong [o]ierr())
Calculate the derivatives of (x,f(x)) using cubic spline interpolation.
Calculate the derivatives, using cubic spline interpolation, at the given points
("$x,$f"), with the specified boundary conditions. Control over the boundary conditions
is given by the $ic and $vc ndarrays. The resulting values - "$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 ndarray $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 ndarray with "2*n" elements.
Error status returned by $ierr:
• 0 if successful.
• -1 if "nelem($x) < 2".
• -3 if $x is not strictly increasing.
• -4 if "ic(0) < 0" or "ic(0) > 4".
• -5 if "ic(1) < 0" or "ic(1) > 4".
• -6 if both of the above are true.
• -7 if "nwk < 2*n".
• -8 in case of trouble solving the linear system for the interior derivative values.
chsp does not process bad values. It will set the bad-value flag of all output ndarrays
if the flag is set for any of the input ndarrays.
chfd
Signature: (x(n);f(n);d(n);longlong check();xe(ne);[o]fe(ne);[o]de(ne);longlong [o]ierr())
Interpolate function and derivative values.
Given a piecewise cubic Hermite function - such as from "chim" - evaluate the function
($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 ndarrays
if the flag is set for any of the input ndarrays.
chfe
Signature: (x(n);f(n);d(n);longlong check();xe(ne);[o]fe(ne);longlong [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 ndarrays
if the flag is set for any of the input ndarrays.
chia
Signature: (x(n);f(n);d(n);longlong check();la();lb();[o]ans();longlong [o]ierr())
Integrate (x,f(x)) over arbitrary limits.
Evaluate the definite integral of a piecewise cubic Hermite function over an arbitrary
interval, given by "[$la,$lb]". $d should contain the derivative values, computed by
"chim". See "chid" if the integration limits are data points. Set "check" to 0 to skip
checks on the input data.
The values of $la and $lb 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 $la lies outside $x.
• 2 if $lb lies outside $x.
• 3 if both 1 and 2 are true.
• -1 if "nelem($x) < 2"
• -3 if $x is not strictly increasing.
• -4 if an error has occurred in a lower-level routine, which should never happen.
chia does not process bad values. It will set the bad-value flag of all output ndarrays
if the flag is set for any of the input ndarrays.
chid
Signature: (x(n);f(n);d(n);longlong check();longlong ia();longlong ib();[o]ans();longlong [o]ierr())
Integrate (x,f(x)) between data points.
Evaluate the definite integral of a a piecewise cubic Hermite function between "x($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. $d should
contain the derivative values, computed by "chim". 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 ndarrays
if the flag is set for any of the input ndarrays.
chcm
Signature: (x(n);f(n);d(n);longlong check();longlong [o]ismon(n);longlong [o]ierr())
Check the given piecewise cubic Hermite function for monotonicity.
The outout ndarray $ismon indicates over which intervals the function is monotonic. Set
"check" to 0 to skip checks on the input data.
For the data interval "[x(i),x(i+1)]", the values of ismon(i) can be:
• -3 if function is probably decreasing
• -1 if function is strictly decreasing
• 0 if function is constant
• 1 if function is strictly increasing
• 2 if function is non-monotonic
• 3 if function is probably increasing
If "abs(ismon(i)) == 3", the derivative values are near the boundary of the monotonicity
region. A small increase produces non-monotonicity, whereas a decrease produces strict
monotonicity.
The above applies to "i = 0 .. nelem($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 ndarrays
if the flag is set for any of the input ndarrays.
chbs
Signature: (x(n);f(n);d(n);longlong knotyp();longlong nknots();t(tsize);[o]bcoef(bsize);longlong [o]ndim();longlong [o]kord();longlong [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 ndarrays
if the flag is set for any of the input ndarrays.
polfit
Signature: (x(n); y(n); w(n); longlong maxdeg(); longlong [o]ndeg(); [o]eps(); [o]r(n); longlong [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 ndarrays if the
flag is set for any of the input ndarrays.
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.