CFD Utilities Files

More than 50 1-dimensional interpolation and smoothing utilities are
gathered in this collection.  Here, interpolation implies passing through
data points precisely, while smoothing methods of course may not.  Many
of the 1-D methods are driven by program SMOOTH, q.v., on one or more
datasets, either to generate usable results or to compare the methods.

Many methods require distinct, monotonic abscissas in the data; others
do not.  Some utilities simply determine and evaluate a certain function
(say a quartic) to fill some interval for given end-point conditions.

Past confusion of (x/y[/z]) methods involving curves in 2- or 3-space
mean some of what are really 1-D methods (single independent variable
such as arc length) appear in the interp2d and interp3d collections,
q.v., and remain there for historical reasons.

The local cubic spline method of LCSFIT and its relatives has proven
extremely valuable, particularly its monotonic option (meaning the
interpolations cannot experience excursions beyond the data points).
Local cubics involve the 4-point Hermite method (cubics defined by two
points and corresponding slopes), with adjustments for monotonicity if
desired. They ensure first derivative continuity at the data points,
but second derivatives may not be continuous.  Nevertheless, even for
geometric applications with smoothly varying data points, the lack of
curvature continuity has proven to be a non-issue.  The curvature-based
point redistribution in CAPSULE_GRID, for example, via the PLSCRV3D
utility (sadly in the interp3d collection, although it belongs here)
as well as LCSFIT, produces excellent surface grids from well-discretized
analytic (or nonanalytic) generatrices.

Note that finite difference derivatives are highly recommended for
calculating curvature -- not splines.  Also, the normalizing and denorm-
alizing done by LCSFIT2 around a call to LCSFIT is recommended unless
the data units are known to be O(1).

See also the aa_nurbs library for application of B-spline curves to
airfoils.

aebxplusc.f90      Nonlinear least squares fit of y = Ae^(B*x) + C, etc.
axbecx.f90         Nonlinear least squares fit of y = Ax^Be^(C*x), etc.
axbplusc.f90       Nonlinear least squares fit of y = Ax^B + C
bernstein.f90      Linear least squares fit of Bernstein basis functions bn(0:n)
cfft.f             Multivariate complex fast Fourier transform
cfft2.f            Bi-directional FFT of a complex sequence
circular_arc.f90   Intended for rounding/filleting vertices in a generatrix
csdval.f           Variant of CSEVAL for f'(x) and f"(x) as well as f(x)
cseval.f           CSFIT companion for evaluating the fitted cubic spline
csfit.f            Compute the coefficients of a conventional cubic spline
cspline.f90        Streamlined interface to CSFIT & CSEVAL, in module form
cubintrp.f         Determine/evaluate cubic defined by 2 points & 2 slopes
fscalc.f           FSCOEF/FSEVAL combination for all-coefs./data pt. evals.
fscoef.f           Fourier coefs. for uniform [0, 2pi] data via recurrence
fsdval.f           FSEVAL variant; evaluates f'(x) & f"(x) as well as f(x)
fseries.f          FSEVAl variant for nonuniform data, full or half period
fseval.f           Evaluate a finite Fourier series, odd | even | neither
fseval2.f          FSEVAL variant for arbitrary units, odd | even | neither
fsfit.f            Fourier coefs. via linear least sqrs.; nonuniform data
fssum.f            Evaluate a finite Fourier series; use w/ FSERIES/FSEVAL2
hermite.f          Evaluate a Hermite cubic on [0, 1] and its derivatives
inverse_parabola.f Variant of QINTRP for upper arm of parabola y^2 = ax + b
lcsfit.f           Local cubic spline for (x,y) data; monotonic option
lcsfit2.f90        Normalizes/denormalizes (x,y) data around an LCSFIT call
lcsinverse.f90     Inverse form of monotonic LCSFIT (x for y in [y2, y3])
legendre.f90       Linear least squares fir of legendre polynomials 0:n
line1d.f           TABLE1 in subroutine form; linear interpln. | extrapoln.
lint1d.f           Multifunction variant of LINE1D
lintrp.f           Variant suited to linear lofting between wing sections
lsfit1.f           Hybrid local llsq. method; smoothed-data spline option
lsfit2.f           F90 upgrade of 1st stage of LSFIT1 local lin. lst. sqrs.
msfit.f            Early monotonic local cubic analogue of CSFIT
nonlinear_interpolation.f LCSFIT + BESSEL/BRODLIE/BUTLAND/THREEPT fns. used
peval2.f           W/ PFIT2, the conventional spline analog of PSFIT/PSEVAL
pfit2.f            Spline coefs. for x = x(t) & y = y(t) on an (x,y) curve
plsfit.f           Original parametric local cubic spline for (x,y) plotting
plsintrp.f         Var. of parametric local cubic: x/y(t); y at xe, not te
pndval.f           PNEVAL variant; evaluates f'(x) & f"(x) as well as f(x)
pneval.f           Evaluation of nth-degree polynomial from PNFIT
pnfit.f            Linear least sqrs. polynomial fit; avoids "normal" eqns.
pnwfit.f           Weighted PNFIT variant; same orthogonal factorzn. method
poly6.f            Deg. 6 polynomial fit for 2 pts. & corresp. y', y"
pseval.f           PSFIT companion: y for given x, not t
psfit.f            Early approach to fitting x, y vs. arc l. t for airfoils
pstval.f           PSFIT companion: y for given t, not x
qhsfit.f           Fit the quasi-Hermite cubic spline of H. Akima, 1970
qinterp.f          Averaged quadratic intrpln. suited to aerodynamic coefs.
qintrp.f           Determine/evaluate a quadratic defined by 2 pts./1 slope
quartic.f          Coefs. of a quartic defined by (x,y,y') and (x,y,y',y")
quartic2.f90       Determines & evaluates a quartic defined as for QUARTIC
quintic.f          Compute coefs. of a quintic between a (x,y,y',y") pair
quintic2.f         Compute quintic defined by (x,y,y') & (x,y,y',y",y"')
radial_basis_1d_weights.f90  Implementation of the radial basis function
radial_basis_1d_eval.f90     interpolation technique described in Wikipedia
rfft2n.f           Fast Fourier coefs. for a sequence of 2n values, n = 2^m
smooth1d.f         Explicit in-place smoothing of a vector; end pts. fixed
smooth1d_niters.f  Variant of SMOOTH1D with a variable number of iterations
supereval.f90      Evaluate indicated superellipse at one or more abscissas
superfit1.f90      Linear least squares portion of fitting a super-ellipse
superfit2.f90      Nonlinear least squares part of fitting a super-ellipse
table1.f           Early 1-D linear interpolation in function form
vecfit.f           Linear least sqrs. fit/eval. of a combination of vectors
wagfit.f           Linear least sqrs. fit of airfoil-like Wagner functions;
                   see also subr. BEVAL used by programs SMOOTH & PROFILE