Slatec(3)

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 "skip"; if set to 1, then no checks on the validity of the input data are made, otherwise these checks are made. The value of "skip" can be set to 1 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", or use "chbs" to convert a PCH function into B-representation for use with the B-spline routines of SLATEC.

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);

  detslatec
compute the determinant of an invertible matrix

  $mat = identity(5); # 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);[t]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: ([io]a(n,n);[o]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: ([io]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: ([io]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: ([io]a(n,n);longlong ipvt(n);[o]det(two=2);[t]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);[io]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();[io]wsave(foo);longlong [o]ifac(ni=15))

Subroutine ezffti initializes the work array "wsave(3n or more)" and ifac() 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);longlong ifac(ni=15))

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);longlong ifac(ni=15))

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.

  polyvalue
  Signature: (longlong l();x();[o]yfit();[o]yp(nder);a(foo); PDL_LongLong nder => nder)

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". Broadcasts correctly over multiple "x" positions.

 ($yfit, $yp) = polyvalue($l, $nder, $x, $c);

polyvalue 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);[t]wk(nwk=CALC(2*$SIZE(n)));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.

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);[t]wk(nwk=CALC(2*$SIZE(n)));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.

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);int [io]skip();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 "skip" 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);int [io]skip();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 "skip" 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);int [io]skip();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 "skip" 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);int [io]skip();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 "skip" 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);int [io]skip();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 "skip" 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 [io]nknots();[io]t(tsize=CALC(2*$SIZE(n)+4));[o]bcoef(bsize=CALC(2*$SIZE(n)));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".

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.

  bvalu
  Signature: (t(nplusk);a(n);longlong ideriv();x();[t]work(k3=CALC(3*($SIZE(nplusk)-$SIZE(n))));[o]ans())

Evaluate the B-representation of a B-spline at X for the function value or any of its derivatives.

Evaluates the B-representation "(T,A,N,K)" of a B-spline at "X" for the function value on "IDERIV = 0" or any of its derivatives on "IDERIV = 1,2,...,K-1". Right limiting values (right derivatives) are returned except at the right end point "X=T(N+1)" where left limiting values are computed. The spline is defined on "T(K) .LE. X .LE. T(N+1)". BVALU returns a fatal error message when "X" is outside of this interval.

To compute left derivatives or left limiting values at a knot T(I), replace "N" by "I-1" and set "X=T(I)", "I=K+1,N+1".

Description of Arguments

Input

     T       - knot vector of length N+K
     A       - B-spline coefficient vector of length N
     N       - number of B-spline coefficients
               N = sum of knot multiplicities-K
     K       - order of the B-spline, K .GE. 1
     IDERIV  - order of the derivative, 0 .LE. IDERIV .LE. K-1
               IDERIV=0 returns the B-spline value
     X       - argument, T(K) .LE. X .LE. T(N+1)
     INBV    - an initialization parameter which must be set
               to 1 the first time BVALU is called. (PDL sets to 1 for you)

Output

     INBV    - INBV contains information for efficient process-
               ing after the initial call and INBV must not
               be changed by the user.  Distinct splines require
               distinct INBV parameters. (unavailable in PDL)
     WORK    - work vector of length 3*K.
     BVALU   - value of the IDERIV-th derivative at X

Error Conditions

An improper input is a fatal error

bvalu 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(); [io]eps(); [o]r(n); longlong [o]ierr(); [o]a(foo=CALC(3*($SIZE(n) + $SIZE(bar)))); [o]coeffs(bar);[t]xtmp(n);[t]ytmp(n);[t]wtmp(n);[t]rtmp(n); IV max_maxdeg_plus1 => bar)

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.