Detailed description of Fitgrid procedures
CURV1 - calculate values for CURV2 usage
This subroutine calculates certain values that are used by
CURV2
in order to compute an interpolatory spline under tension through
a sequence of functional values. The actual computation of the
interpolated values must be done using CURV2.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL CURV1 (N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
SLP1, | Real | Input |
SLPN, | Real | Input |
ISLPSW, | Integer | Input |
YP, | Real | Output | N
TEMP, | Real | Input | N
SIGMA, | Real | Input |
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input
data -- (Y(K) is the functional value at X(K) for K=1,N).
- SLP1
- A user-specified value for the desired slope at X(1). See
ISLPSW below if you want to have a value calculated internally.
- SLPN
- A user-specified value for the desired slope at X(N). See
ISLPSW below if you want to have a value calculated internally.
- ISLPSW
- A switch to indicate whether the slopes at the end points should
be calculated internally.
- = 0 if SLP1 and SLPN are user-specified.
- = 1 if SLP1 is user-specified, but SLPN is internally calculated.
- = 2 if SLPN is user-specified, but SLP1 is internally calculated.
- = 3 if SLP1 and SLPN are internally calculated.
- YP
- Contains values for the second derivative (CURV1 computes these).
- TEMP
- Scratch space.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if X values are not strictly increasing.
CURV2 - interpolate at a specified point
This function interpolates a value at a specified point using
a spline under tension.
CURV1 must be called prior to calling CURV2,
and the values thus obtained used as input to CURV2.
CURV2 returns the interpolated function value at the X-coordinate specified.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
FUNCTION CURV2 (T, | Real | Input |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
YP, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- T
- The abscissa for which an interpolated function value is desired.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input data --
Y(K) is the functional value at X(K) for K=1,N.
- YP
- Contains values for the second derivative (as calculated by
CURV1).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVD - calculate a derivative at a specified point
This function calculates a derivative at a specified point using
a spline under tension.
CURV1 must be called prior to calling CURVD,
and the values thus obtained used as input to CURVD.
The derivative value is returned.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
FUNCTION CURVD (T, | Real | Input |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
YP, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- T
- The abscissa for which an interpolated function value is desired.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- YP
- Contains values for the second derivative (as calculated by
CURV1).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVI - integrate a spline curve
This function calculates an integral between two specified limits.
CURV1 must be called prior to calling CURVI,
and the values thus obtained used as input to CURVI.
The value of the integral is returned.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
FUNCTION CURVI (XL, | Real | Input |
XU, | Real | Input |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
YP, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- XL
- The lower limit of the integration.
- XR
- The upper limit of the integration.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- YP
- Contains values for the second derivative (as calculated
by CURV1).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVP1 - calculate values for CURVP2 usage
This subroutine calculates certain values that are used by
CURVP2
in order to compute an interpolatory spline under tension through
a sequence of functional values for a periodic function.
The actual computation of the
interpolated values must be done using CURVP2.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL CURVP1 (N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
P, | Real | Input |
YP, | Real | Output | N
TEMP, | Real | Input | N x 2
SIGMA, | Real | Input |
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- P
- The period of the function. P must be greater than
X(N)-X(1).
- YP
- Contains values for the second derivative (these are calculated by
CURVP1).
- TEMP
- Scratch space.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if P is less than or equal to X(N)-X(1).
- = 3 if the X values are not strictly increasing.
CURVP2 - interpolate a periodic function at a specified point
This function interpolates a value at a specified point using
a spline under tension.
CURVP1 must be called prior to calling CURVP2,
and the values thus obtained used as input to CURVP2.
The interpolated value is returned.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
FUNCTION CURVP2 (T, | Real | Input |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
P, | Real | Input |
YP, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- T
- The abscissa for which an interpolated function value is desired.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values for the input function --
Y(K) is the functional value at X(K) for K=1,N.
- P
- The period of the function.
- YP
- Contains values for the second derivative (as computed by CURVP1).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVPI - integrate a periodic spline curve
This function calculates an integral between two specified limits.
CURVP1 must be called prior to calling CURVPI,
and the values thus obtained used as input to CURVPI.
The value of the integral is returned.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
FUNCTION CURVI (XL, | Real | Input |
XU, | Real | Input |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
P, | Real | Input |
YP, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- XL
- The lower limit of the integration.
- XR
- The upper limit of the integration.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- P
- The period of the function.
- YP
- Contains values for the second derivative (as calculated by
CURVP1).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVS - calculate values for a smoothing spline
This subroutine calculates certain values that are used by
CURV2
in order to compute an interpolatory smoothing spline under tension through
a sequence of functional values. The actual computation of the
interpolated values must be done using CURV2.
Three parameters are used to control the degree of smoothness -- D, S,
and EPS.
The parameter D is a value indicating the degree of confidence
in the accuracy of the input function values -- it should be an
approximation of the standard deviation of error. Effectively the value
of D controls how close the smoothed curve comes to the input data
points. If D is small then the interpolated curve will pass close to the
input data. The larger the value of D, the more freedom the smooth
curve has in how close it comes to the input data values.
S is a
more subtle global smoothing parameter. S must be non-negative. For small
values of S, the interpolated curve approximates the tension
spline and for larger
values of S, the curve is smoother. A reasonable value for EPS is
REAL(N).
EPS controls the precision to which S is interpreted; EPS
must be between 0. and 1. inclusive. A reasonable value for S is
SQRT(2./REAL(N)).
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL CURVS (N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
D, | Real | Input | N
ISW, | Integer | Input |
S, | Real | Input |
EPS, | Real | Input |
YS, | Real | Output | N
YSP, | Real | Output | N
SIGMA, | Real | Input |
TEMP, | Real | Input | N x 9
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values for the input function --
Y(K) is the functional value at X(K) for K=1,N.
- D
- A user-specified value containing the observed weights. D may
be either an array or a scalar, depending on the value of ISW
(as described below).
- ISW
- A switch for interpreting the value of D. If ISW=0, then D is
an array of length N (D contains an individual error estimate for each
input data value);
if ISW=1, then D is a scalar that serves as an error estimate
for every single data item.
- S
- Contains the value for smoothing. S must be non-negative. Larger
values for S yield greater smoothing. A reasonable value is REAL(N).
- EPS
- Contains a tolerance value for the relative precision to which
S should be interpreted. EPS must be between 0. and 1. inclusive.
A reasonable value is SQRT(2./REAL(N)).
- YS
- An array of length N.
- YSP
- An array of length N.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- TEMP
- Scratch space.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if S is negative.
- = 3 if EPS is negative or greater than 1.
- = 4 if X values are not strictly increasing.
- = 5 if D is negative.
CURVS1 - calculate values for a smoothing spline in the plane to be used by CURVS2
This subroutine calculates certain values that are used by
CURVS2
in order to compute an interpolatory smoothing spline under tension through
a sequence of functional values in the plane. The actual computation of the
interpolated values must be done using CURVS2.
Three parameters are used to control the degree of smoothness -- D, S,
and EPS.
The parameter D is a value indicating the degree of confidence
in the accuracy of the input function values -- it should be an
approximation of the standard deviation of error. Effectively the value
of D controls how close the smoothed curve comes to the input data
points. If D is small then the interpolated curve will pass close to the
input data. The larger the value of D, the more freedom the smooth
curve has in how close it comes to the input data values.
S is a
more subtle global smoothing parameter. S must be non-negative. For small
values of S, the interpolated curve approximates the tension
spline and for larger
values of S, the curve is smoother. A reasonable value for EPS is
REAL(N).
EPS controls the precision to which S is interpreted; EPS
must be between 0. and 1. inclusive. A reasonable value for S is
SQRT(2./REAL(N)).
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL CURVS1 (N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
D, | Real | Input | N or 1 (see desc.)
ISW, | Integer | Input |
S, | Real | Input |
EPS, | Real | Input |
PARAM, | Real | Output | N
XS, | Real | Output | N
YS, | Real | Output | N
XSP, | Real | Output | N
YSP, | Real | Output | N
SIGMA, | Real | Input |
TEMP, | Real | Input | N x 19
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the X-coordinates for the input
data. These need not be increasing.
- Y
- An array containing the Y-coordinates for the input
data.
- D
- A user-specified value containing the observed weights. D may
be either an array or a scalar, depending on the value of ISW
(as described below). D effectively controls how close the
interpolated curve will come to the input points.
- ISW
- A switch for interpreting the value of D. If ISW=0, then D is
an array of length N (D contains an individual error estimate for each
input data value);
if ISW=1, then D is a scalar that serves as an error estimate
for every single data item.
- S
- Contains the value for smoothing. S must be non-negative. Larger
values for S yield greater smoothing. A reasonable value is REAL(N).
- EPS
- Contains a tolerance value for the relative precision to which
S should be interpreted. EPS must be between 0. and 1. inclusive.
A reasonable value is SQRT(2./REAL(N)).
- XS
- An array of length N, contains the smoothed values.
- YS
- An array of length N, contains the smoothed values.
- XSP
- An array of length N. Contains second derivative
information for the X-coordinate values.
- YSP
- An array of length N. Contains second derivative
information for the Y-coordinate values.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- TEMP
- Scratch space.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if S is negative.
- = 3 if EPS is negative or greater than 1.
- = 4 if X values are not strictly increasing.
- = 5 if D is negative.
CURVS2 - calculate values for a smoothing spline in the plane
This subroutine is a companion to CURV1S which must be
called before calling this subroutine. CURV2S calculates an
interpolated value for a smoothing spline under tension at a
given point.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL CURVS2 (T, | Real | Input |
N, | Integer | Input |
PARAM, | Real | Input | N
X, | Real | Input | N
Y, | Real | Input | N
XP, | Real | Input | N
YP, | Real | Input | N
SIGMA, | Real | Input |
XO, | Real | Output |
YO) | Real | Output |
--------------------------------------------------------------
- T
- A real value to be mapped onto the
interpolating curve. Values of T between zero
and one interpolate the original data; any
values of T outside this range result in extrapolation.
- N
- The number of points which were specified to determine
the curve in CURVS1.
- PARAM
- The arc lengths as computed by CURVS1.
- X
- An array containing the X-coordinates for the input
data as input to CURVS1. These need not be increasing.
- Y
- An array containing the Y-coordinates for the input data as
input to CURVS1.
- XP
- An array of second derivative values as calculated by CURVS1.
- YP
- An array of second derivative values as calculated by CURVS1.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- XO
- The interpolated X value.
- YO
- The interpolated Y value.
CURVPS - calculate values for a smoothing spline
This subroutine calculates certain values that are used by
CURVP2
in order to compute an interpolatory smoothing spline under tension through
a sequence of functional values for a periodic function.
The actual computation of the
interpolated values must be done using CURVP2.
Three parameters are used to control the degree of smoothness -- D, S,
and EPS.
The parameter D is a value indicating the degree of confidence
in the accuracy of the input function values -- it should be an
approximation of the standard deviation of error. Effectively the value
of D controls how close the smoothed curve comes to the input data
points. If D is small then the interpolated curve will pass close to the
input data. The larger the value of D, the more freedom the smooth
curve has in how close it comes to the input data values.
S is a
more subtle global smoothing parameter. S must be non-negative. For small
values of S, the curve approximates the tension spline and for larger
values of S, the curve is smoother. A reasonable value for S is
REAL(N).
EPS controls the precision to which S is interpreted; EPS
must be between 0. and 1. inclusive. A reasonable value for EPS is
SQRT(2./REAL(N)).
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL CURVPS (N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
P, | Real | Input |
D, | Real | Input | N
ISW, | Integer | Input |
S, | Real | Input |
EPS, | Real | Input |
YS, | Real | Output | N
YSP, | Real | Output | N
SIGMA, | Real | Input |
TEMP, | Real | Input | N x 11
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- P
- The period of the function. P must be greater than
X(N)-X(1).
- D
- A user-specified value containing the observed weights. D may
either be an array or a scalar, depending on the value of ISW.
- ISW
- A switch for interpreting the value of D. If ISW=0, then D is
an array of length N (an error estimate for each input data value);
if ISW=1, then D is a scalar that serves as an error estimate
for every single data item.
- S
- Contains the value for smoothing. S must be non-negative. Larger
values for S yield greater smoothing. A reasonable value is REAL(N).
- EPS
- Contains a tolerance value for the relative precision to which
S should be interpreted. EPS must be between 0. and 1. inclusive.
A reasonable value is SQRT(2./REAL(N)).
- YS
- An array of length N.
- YSP
- An array of length N.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- TEMP
- Scratch space.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if S is negative.
- = 3 if EPS is negative or greater than 1.
- = 4 if X values are not strictly increasing.
- = 5 if D is negative.
- = 6 if P is less than or equal to X(N)-X(1).
KURV1 - calculate values for KURV2 usage
This subroutine calculates certain values that are used by
KURV2
in order to compute an interpolatory spline under tension through
a sequence of points in the plane. The actual computation of the
interpolated values must be done using KURV2.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURV1 (N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
SLP1, | Real | Input |
SLPN, | Real | Input |
ISLPSW, | Integer | Input |
XP, | Real | Output | N
YP, | Real | Output | N
TEMP, | Real | Input | N
S, | Real | Output | N
SIGMA, | Real | Input |
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the X values of the points.
- Y
- An array containing the Y values of the points.
Adjacent pairs of points must be distinct.
- SLP1
- A value (in radians) containing the slope at (X(1),Y(1)). The
angles are measured counter-clockwise from the X axis and the
positive sense of the curve is assumed to be that moving from
point 1 to point N. A value for SLP1 may be omitted as indicated
by the switch ISLPSW.
- SLPN
- A value (in radians) containing the slope at (X(N),Y(N)). The
angles are measured counter-clockwise from the X axis and the
positive sense of the curve is assumed to be that moving from
point 1 to point N. A value for SLP2 may be omitted as indicated
by the switch ISLPSW.
- ISLPSW
- A switch to indicate whether the slopes at the end points should
be calculated internally.
- = 0 if SLP1 and SLPN are user-specified.
- = 1 if SLP1 is user-specified, but SLPN calculated internally.
- = 2 if SLPN is user-specified, but SLP1 calculated internally.
- = 3 if SLP1 and SLPN are internally calculated.
- XP
- An array of length N.
- YP
- An array of length N.
- TEMP
- Scratch space.
- S
- An array of length N. On output, S contains the polygonal arclengths
of the curve.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if a pair of adjacent points coincide.
KURV2 - does interpolation for parametric curves
The interpolated curve is parameterized by mapping
points in the interval [0.,1.] onto the interpolated curve.
The resulting curve has a parametric representation both
of whose components are splines under tension and functions of the
polygonal arc length. The value 0. is mapped onto (X[1],Y[1]) and
the value 1. is mapped onto (X[N],Y[N]).
KURV1 must be called before calling KURV2.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURV2 (T, | Real | Input |
XS, | Real | Output |
YS, | Real | Output |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
XP, | Real | Input | N
YP, | Real | Input | N
S, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- T
- A value to be mapped to a point on the curve. The interval
[0.,1.] is mapped onto the curve such that 0. is mapped to
(X(1),Y(1)) and 1. is mapped to (X(N),Y(N)). Values outside
of [0.,1.] result in extrapolation.
- XS
- Contains the X coordinate of the interpolated point that T maps to.
- YS
- Contains the Y coordinate of the interpolated point that T maps to.
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the input points.
- Y
- An array containing the Y values of the input points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N as computed by KURV1.
- YP
- An array of length N as computed by KURV1.
- S
- An array of length N containing the polygonal arc lengths of the
curve.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
KURVD - interpolation and derivatives for parametric curves
This subroutine behaves like KURV2 except that in
addition it returns the first and second derivatives of the component
functions in the parameterization. The interval [0.,1.] maps onto a curve in
the plane. The resulting curve has a parametric representation both
of whose components are splines under tension and functions of the
polygonal arc length.
KURV1 must be called before calling KURVD.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURVD (T, | Real | Input |
XS, | Real | Output |
YS, | Real | Output |
XST, | Real | Output |
YST, | Real | Output |
XSTT, | Real | Output |
YSTT, | Real | Output |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
XP, | Real | Input | N
YP, | Real | Input | N
S, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- T
- A value to be mapped to a point on the curve. The interval
[0.,1.] is mapped onto the curve such that 0. is mapped to
(X(1),Y(1)) and 1. is mapped to (X(N),Y(N)). Values outside
of [0.,1.] result in extrapolation.
- XS
- Contains the X coordinate of the point that T maps to.
- YS
- Contains the Y coordinate of the point that T maps to.
- XST
- Contains the first derivatives of the X component with
respect to T.
- YST
- Contains the first derivatives of the Y component with
respect to T.
- XSTT
- Contains the second derivatives of the X component with
respect to T.
- YSTT
- Contains the second derivatives of the X component with
respect to T.
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the input points.
- Y
- An array containing the Y values of the input points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N as computed by KURV1.
- YP
- An array of length N as computed by KURV1.
- S
- An array of length N containing the polygonal arc lengths of the
curve, as computed by KURV1.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
KURVP1 - calculate values for KURVP2 usage
This subroutine calculates certain values that are used by
KURVP2
in order to compute an interpolatory spline under tension through
a sequence of points in the plane that form a closed curve.
The actual computation of the
interpolated values must be done using KURVP2.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURVP1 (N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
XP, | Real | Output | N
YP, | Real | Output | N
TEMP, | Real | Input | N x 2
S, | Real | Output | N
SIGMA, | Real | Input |
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the points.
- Y
- An array containing the Y values of the points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N.
- YP
- An array of length N.
- TEMP
- Scratch space.
- S
- An array of length N.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if a pair of adjacent points coincide.
KURVP2 - does interpolation for closed
parametric curves
The interpolated curve is parameterized by mapping
points in the interval [0.,1.] onto an interpolated closed curve.
The resulting curve has a parametric representation both
of whose components are splines under tension and functions of the
polygonal arc length. The value 0. is mapped onto (X[1],Y[1]) and
the value 1. is also mapped onto (X[1],Y[1]), since the interpolated
curve is closed.
KURVP1 must be called before calling KURVP2.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURVP2 (T, | Real | Input |
XS, | Real | Output |
YS, | Real | Output |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
XP, | Real | Input | N
YP, | Real | Input | N
S, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- T
- A value to be mapped to a point on the curve. The interval
[0.,1.] is mapped onto the curve such that 0. is mapped to
(X(1),Y(1)) and 1. is mapped to (X(1),Y(1)), since the
interpolated curve is closed. Any interval [TT,TT+1.]
maps onto the entire curve.
- XS
- Contains the X coordinate of the point that T maps to.
- YS
- Contains the Y coordinate of the point that T maps to.
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the points.
- Y
- An array containing the Y values of the points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N as computed by KURV1.
- YP
- An array of length N as computed by KURV1.
- S
- An array of length N containing the polygonal arc lengths of the
curve, as computed by KURV1.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
KURVPD - interpolation and derivatives for closed parametric curves
This subroutine behaves like KURVP2 except that in
addition it returns the first and second derivatives of the component
functions in the parameterization. The interval [0.,1.] maps onto a closed
curve in
the plane. The resulting curve has a parametric representation both
of whose components are splines under tension and functions of the
polygonal arc length.
KURVP1 must be called before calling KURVPD.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURVPD (T, | Real | Input |
XS, | Real | Output |
YS, | Real | Output |
XST, | Real | Output |
YST, | Real | Output |
XSTT, | Real | Output |
YSTT, | Real | Output |
N, | Integer | Input |
X, | Real | Input | N
Y, | Real | Input | N
XP, | Real | Input | N
YP, | Real | Input | N
S, | Real | Input | N
SIGMA) | Real | Input |
--------------------------------------------------------------
- T
- A value to be mapped to a point on the curve. The interval
[0.,1.] is mapped onto the curve such that 0. is mapped to
(X(1),Y(1)) and 1. is mapped to (X(1),Y(1)). Any interval
[TT,TT+1.] maps onto the entire closed curve.
- XS
- Contains the X coordinate of the point that T maps to.
- YS
- Contains the Y coordinate of the point that T maps to.
- XST
- Contains the first derivatives of the X component with
respect to T.
- YST
- Contains the first derivatives of the Y component with
respect to T.
- XSTT
- Contains the second derivatives of the X component with
respect to T.
- YSTT
- Contains the second derivatives of the X component with
respect to T.
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the input points.
- Y
- An array containing the Y values of the input points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N as computed by KURVP1.
- YP
- An array of length N as computed by KURVP1.
- S
- An array of length N containing the polygonal arc lengths of the
curve, as computed by KURVP1.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
SURF1 - calculate values for SURF2 usage
This subroutine calculates certain values that are used by
SURF2
in order to compute an interpolatory surface passing through a
rectangular grid of function values. The surface computed is a
tensor product of splines under tension. To calculate actual
interpolated values, SURF2 must be called.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL SURF1 (M, | Integer | Input |
N, | Integer | Input |
X, | Real | Input | M
Y, | Real | Input | N
Z, | Real | Input | IZ x N
IZ, | Integer | Input |
ZX1, | Real | Input | N
ZXM, | Real | Input | N
ZY1, | Real | Input | M
ZYN, | Real | Input | M
ZXY11, | Real | Input |
ZXYM1, | Real | Input |
ZXY1N, | Real | Input |
ZXYMN, | Real | Input |
ISLPSW, | Integer | Input |
ZP, | Real | Output | M x N x 3
TEMP, | Real | Input | N + N + M
SIGMA, | Real | Input |
IER) | Integer | Output |
--------------------------------------------------------------
- M
- The number of grid lines in the X direction. (M > 1)
- N
- The number of grid lines in the Y direction. (N > 1)
- X
- An array containing M X coordinates for grid lines in the X direction.
These values must be strictly increasing.
- Y
- An array containing N Y coordinates for grid lines in the Y direction.
These values must be strictly increasing.
- Z
- An array containing M x N functional values at the grid points;
Z(I,J) contains the functional value at (X(I),Y(J)) for I=1,M and
J=1,N.
- IZ
- The row dimension of the matrix Z (IZ is greater than or equal to M).
- ZX1
- An array containing N X-partial derivatives of the function along
the line X(1), that is ZX1(J) is the X-partial derivative at point
(X(1),Y(J)) for J=1,N. This parameter may be defaulted by setting
ISLPSW appropriately.
- ZXM
- An array containing N X-partial derivatives of the function along
the line X(M), that is ZXM(J) is the X-partial derivative at point
(X(M),Y(J)) for J=1,N. This parameter may be defaulted by setting
ISLPSW appropriately.
- ZY1
- An array containing M Y-partial derivatives of the function along
the line Y(1), that is ZY1(I) is the Y-partial derivative at point
(X(I),Y(1)) for I=1,M. This parameter may be defaulted by setting
ISLPSW appropriately.
- ZYN
- An array containing M Y-partial derivatives of the function along
the line Y(N), that is ZY1(I) is the Y-partial derivative at point
(X(I),Y(N)) for I=1,M. This parameter may be defaulted by setting
ISLPSW appropriately.
- ZXY11
- The X-Y-partial derivative at (X(1),Y(1). This parameter may
be defaulted by setting ISLPSW appropriately.
- ZXYM1
- The X-Y-partial derivative at (X(M),Y(1). This parameter may
be defaulted by setting ISLPSW appropriately.
- ZXY1N
- The X-Y-partial derivative at (X(1),Y(N). This parameter may
be defaulted by setting ISLPSW appropriately.
- ZXYMN
- The X-Y-partial derivative at (X(M),Y(N). This parameter may
be defaulted by setting ISLPSW appropriately.
- ISLPSW
- A switch to indicate which boundary derivatives are user
supplied and which should be estimated internally. Where
- I1 = 0 if ZX1 is user-supplied and 1 otherwise.
- I2 = 0 if ZXM is user-supplied and 1 otherwise.
- I3 = 0 if ZY1 is user-supplied and 1 otherwise.
- I4 = 0 if ZYN is user-supplied and 1 otherwise.
- I5 = 0 if ZXY11 is user-supplied and 1 otherwise.
- I6 = 0 if ZXYM1 is user-supplied and 1 otherwise.
- I7 = 0 if ZXY1N is user-supplied and 1 otherwise.
- I8 = 0 if ZXYMN is user-supplied and 1 otherwise.
set ISLPSW = I1 + 2*I2 + 4*I3 +8*I4 +16*I5 +32*I6 + 64*I7 + 128*I8.
Then, for example, if ISLPSW=0, then all derivative information
is user-supplied and if
ISLPSW=255, then all derivative information is to be internally
estimated.
- ZP
- An array of size M x N x 3. On output this contains partial
derivatives of the surface at the given nodes. ZP is used by
SURF2.
- TEMP
- Scratch space.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if X or Y values are not strictly increasing.
SURF2 - 2D interpolation for gridded data
This function interpolates a surface value at a specified coordinate
using bi-splines under tension.
SURF1 must be called before invoking SURF2.
The desired interpolated value is returned as the value of the function.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
FUNCTION SURF2 (XX, | Real | Input |
YY, | Real | Input |
M, | Integer | Input |
N, | Integer | Input |
X, | Real | Input | M
Y, | Real | Input | N
Z, | Real | Input | IZ x N
IZ, | Integer | Input |
ZP, | Real | Input | M x N x 3
SIGMA) | Real | Input |
--------------------------------------------------------------
- XX
- Contains the X coordinate of a point to be mapped onto the interpolated
surface.
- YY
- Contains the Y coordinate of a point to be mapped onto the interpolated
surface.
- M
- The number of grid lines in the X direction. (M > 1)
- N
- The number of grid lines in the Y direction. (N > 1)
- X
- An array containing M X coordinates for grid lines in the X direction.
These values must be strictly increasing.
- Y
- An array containing N Y coordinates for grid lines in the Y direction.
These values must be strictly increasing.
- Z
- An array containing M x N functional values at the grid points;
Z(I,J) contains the functional value at (X(I),Y(J)) for I=1,M and
J=1,N.
- IZ
- The row dimension of the matrix Z (IZ is greater than or equal to M).
- ZP
- An array of size M x N x 3.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURV1DP - calculate values for CURV2DP
usage
(Double precision version of CURV1)
This subroutine calculates certain values that are used by CURV2DP in order to compute an interpolatory
spline under tension through a sequence of functional values. The
actual computation of the interpolated values must be done using CURV2DP.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL CURV1DP (N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
SLP1, | Double precision | Input |
SLPN, | Double precision | Input |
ISLPSW, | Integer | Input |
YP, | Double precision | Output | N
TEMP, | Double precision | Input | N
SIGMA, | Double precision | Input |
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input
data -- (Y(K) is the functional value at X(K) for K=1,N).
- SLP1
- A user-specified value for the desired slope at X(1). See
ISLPSW below if you want to have a value calculated internally.
- SLPN
- A user-specified value for the desired slope at X(N). See
ISLPSW below if you want to have a value calculated internally.
- ISLPSW
- A switch to indicate whether the slopes at the end points should
be calculated internally.
- = 0 if SLP1 and SLPN are user-specified.
- = 1 if SLP1 is user-specified, but SLPN is internally calculated.
- = 2 if SLPN is user-specified, but SLP1 is internally calculated.
- = 3 if SLP1 and SLPN are internally calculated.
- YP
- Contains values for the second derivative (CURV1DP computes these).
- TEMP
- Scratch space.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if X values are not strictly increasing.
CURV2DP - interpolate at a specified point
(Double precision version of CURV2)
This function interpolates a value at a specified point using a spline
under tension. CURV1DP must be called prior to
calling CURV2DP, and the values thus obtained used as input to
CURV2DP. CURV2DP returns the interpolated function value at the
X-coordinate specified.
------------------------------------------------------------------
Argument | Type | Mode | Dimension
------------------------------------------------------------------
FUNCTION CURV2DP (T, | Double precision | Input |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
YP, | Double precision | Input | N
SIGMA) | Double precision | Input |
------------------------------------------------------------------
- T
- The abscissa for which an interpolated function value is desired.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input data --
Y(K) is the functional value at X(K) for K=1,N.
- YP
- Contains values for the second derivative (as calculated by
CURV1DP).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVDDP - calculate a derivative at a specified point
(Double precision version of CURVD)
This function calculates a derivative at a specified point using a
spline under tension. CURV1DP must be called
prior to calling CURVDDP, and the values thus obtained used as input to
CURVDDP. The derivative value is returned.
------------------------------------------------------------------
Argument | Type | Mode | Dimension
------------------------------------------------------------------
FUNCTION CURVDDP (T, | Double precision | Input |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
YP, | Double precision | Input | N
SIGMA) | Double precision | Input |
------------------------------------------------------------------
- T
- The abscissa for which an interpolated function value is desired.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- YP
- Contains values for the second derivative (as calculated by
CURV1DP).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVIDP - integrate a spline curve
(Double precision version of CURVI)
This function calculates an integral between two specified limits. CURV1DP must be called prior to calling CURVIDP,
and the values thus obtained used as input to CURVIDP. The value of
the integral is returned.
------------------------------------------------------------------
Argument | Type | Mode | Dimension
------------------------------------------------------------------
FUNCTION CURVIDP (XL, | Double precision | Input |
XU, | Double precision | Input |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
YP, | Double precision | Input | N
SIGMA) | Double precision | Input |
------------------------------------------------------------------
- XL
- The lower limit of the integration.
- XR
- The upper limit of the integration.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- YP
- Contains values for the second derivative (as calculated
by CURV1DP).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVP1DP - calculate values for CURVP2DP usage
(Double precision version of CURVP1)
This subroutine calculates certain values that are used by CURVP2DP in order to compute an interpolatory
spline under tension through a sequence of functional values for a
periodic function. The actual computation of the interpolated values
must be done using CURVP2DP.
---------------------------------------------------------------
Argument | Type | Mode | Dimension
---------------------------------------------------------------
CALL CURVP1DP (N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
P, | Double precision | Input |
YP, | Double precision | Output | N
TEMP, | Double precision | Input | N x 2
SIGMA, | Double precision | Input |
IER) | Integer | Output |
---------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- P
- The period of the function. P must be greater than
X(N)-X(1).
- YP
- Contains values for the second derivative (these are calculated by
CURVP1DP).
- TEMP
- Scratch space.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if P is less than or equal to X(N)-X(1).
- = 3 if the X values are not strictly increasing.
CURVP2DP - interpolate a periodic function at a specified point
(Double precision version of CURVP2)
This function interpolates a value at a specified point using a spline
under tension. CURVP1DP must be called prior
to calling CURVP2DP, and the values thus obtained used as input to
CURVP2DP. The interpolated value is returned.
-------------------------------------------------------------------
Argument | Type | Mode | Dimension
-------------------------------------------------------------------
FUNCTION CURVP2DP (T, | Double precision | Input |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
P, | Double precision | Input |
YP, | Double precision | Input | N
SIGMA) | Double precision | Input |
-------------------------------------------------------------------
- T
- The abscissa for which an interpolated function value is desired.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values for the input function --
Y(K) is the functional value at X(K) for K=1,N.
- P
- The period of the function.
- YP
- Contains values for the second derivative (as computed by CURVP1DP).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVPIDP - integrate a periodic spline curve
(Double precision version of CURVPI)
This function calculates an integral between two specified limits. CURVP1DP must be called prior to calling CURVPIDP,
and the values thus obtained used as input to CURVPIDP. The value of
the integral is returned.
------------------------------------------------------------------
Argument | Type | Mode | Dimension
------------------------------------------------------------------
FUNCTION CURVIDP (XL, | Double precision | Input |
XU, | Double precision | Input |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
P, | Double precision | Input |
YP, | Double precision | Input | N
SIGMA) | Double precision | Input |
------------------------------------------------------------------
- XL
- The lower limit of the integration.
- XR
- The upper limit of the integration.
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- P
- The period of the function.
- YP
- Contains values for the second derivative (as calculated by
CURVP1DP).
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
CURVSDP - calculate values for a smoothing spline
(Double precision version of CURVS)
This subroutine calculates certain values that are used by
CURV2DP
in order to compute an interpolatory smoothing spline under tension through
a sequence of functional values. The actual computation of the
interpolated values must be done using CURV2DP.
Three parameters are used to control the degree of smoothness -- D, S,
and EPS.
The parameter D is a value indicating the degree of confidence
in the accuracy of the input function values -- it should be an
approximation of the standard deviation of error. Effectively the value
of D controls how close the smoothed curve comes to the input data
points. If D is small then the interpolated curve will pass close to the
input data. The larger the value of D, the more freedom the smooth
curve has in how close it comes to the input data values.
S is a
more subtle global smoothing parameter. S must be non-negative. For small
values of S, the interpolated curve approximates the tension
spline and for larger
values of S, the curve is smoother. A reasonable value for EPS is
REAL(N).
EPS controls the precision to which S is interpreted; EPS
must be between 0. and 1. inclusive. A reasonable value for S is
SQRT(2./REAL(N)).
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL CURVSDP (N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
D, | Double precision | Input | N
ISW, | Integer | Input |
S, | Double precision | Input |
EPS, | Double precision | Input |
YS, | Double precision | Output | N
YSP, | Double precision | Output | N
SIGMA, | Double precision | Input |
TEMP, | Double precision | Input | N x 9
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values for the input function --
Y(K) is the functional value at X(K) for K=1,N.
- D
- A user-specified value containing the observed weights. D may
be either an array or a scalar, depending on the value of ISW
(as described below).
- ISW
- A switch for interpreting the value of D. If ISW=0, then D is
an array of length N (D contains an individual error estimate for each
input data value);
if ISW=1, then D is a scalar that serves as an error estimate
for every single data item.
- S
- Contains the value for smoothing. S must be non-negative. Larger
values for S yield greater smoothing. A reasonable value is REAL(N).
- EPS
- Contains a tolerance value for the relative precision to which
S should be interpreted. EPS must be between 0. and 1. inclusive.
A reasonable value is SQRT(2./REAL(N)).
- YS
- An array of length N.
- YSP
- An array of length N.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- TEMP
- Scratch space.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if S is negative.
- = 3 if EPS is negative or greater than 1.
- = 4 if X values are not strictly increasing.
- = 5 if D is negative.
CURVS1DP - calculate values for a smoothing spline in the plane to be used by CURVS2DP
(Double precision version of CURVS1)
This subroutine calculates certain values that are used by
CURVS2DP
in order to compute an interpolatory smoothing spline under tension through
a sequence of functional values in the plane. The actual computation of the
interpolated values must be done using CURVS2DP.
Three parameters are used to control the degree of smoothness -- D, S,
and EPS.
The parameter D is a value indicating the degree of confidence
in the accuracy of the input function values -- it should be an
approximation of the standard deviation of error. Effectively the value
of D controls how close the smoothed curve comes to the input data
points. If D is small then the interpolated curve will pass close to the
input data. The larger the value of D, the more freedom the smooth
curve has in how close it comes to the input data values.
S is a
more subtle global smoothing parameter. S must be non-negative. For small
values of S, the interpolated curve approximates the tension
spline and for larger
values of S, the curve is smoother. A reasonable value for EPS is
REAL(N).
EPS controls the precision to which S is interpreted; EPS
must be between 0. and 1. inclusive. A reasonable value for S is
SQRT(2./REAL(N)).
---------------------------------------------------------------
Argument | Type | Mode | Dimension
---------------------------------------------------------------
CALL CURVS1DP (N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
D, | Double precision | Input | N or 1 (see desc.)
ISW, | Integer | Input |
S, | Double precision | Input |
EPS, | Double precision | Input |
PARAM, | Double precision | Output | N
XS, | Double precision | Output | N
YS, | Double precision | Output | N
XSP, | Double precision | Output | N
YSP, | Double precision | Output | N
SIGMA, | Double precision | Input |
TEMP, | Double precision | Input | N x 19
IER) | Integer | Output |
---------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the X-coordinates for the input
data. These need not be increasing.
- Y
- An array containing the Y-coordinates for the input
data.
- D
- A user-specified value containing the observed weights. D may
be either an array or a scalar, depending on the value of ISW
(as described below). D effectively controls how close the
interpolated curve will come to the input points.
- ISW
- A switch for interpreting the value of D. If ISW=0, then D is
an array of length N (D contains an individual error estimate for each
input data value);
if ISW=1, then D is a scalar that serves as an error estimate
for every single data item.
- S
- Contains the value for smoothing. S must be non-negative. Larger
values for S yield greater smoothing. A reasonable value is REAL(N).
- EPS
- Contains a tolerance value for the relative precision to which
S should be interpreted. EPS must be between 0. and 1. inclusive.
A reasonable value is SQRT(2./REAL(N)).
- XS
- An array of length N, contains the smoothed values.
- YS
- An array of length N, contains the smoothed values.
- XSP
- An array of length N. Contains second derivative
information for the X-coordinate values.
- YSP
- An array of length N. Contains second derivative
information for the Y-coordinate values.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- TEMP
- Scratch space.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if S is negative.
- = 3 if EPS is negative or greater than 1.
- = 4 if X values are not strictly increasing.
- = 5 if D is negative.
CURVS2DP - calculate values for a smoothing spline in the plane
(Double precision version of CURVS2)
This subroutine is a companion to CURV1SDP which must be called before
calling this subroutine. CURV2SDP calculates an interpolated value
for a smoothing spline under tension at a given point.
---------------------------------------------------------------
Argument | Type | Mode | Dimension
---------------------------------------------------------------
CALL CURVS2DP (T, | Double precision | Input |
N, | Integer | Input |
PARAM, | Double precision | Input | N
X, | Double precision | Input | N
Y, | Double precision | Input | N
XP, | Double precision | Input | N
YP, | Double precision | Input | N
SIGMA, | Double precision | Input |
XO, | Double precision | Output |
YO) | Double precision | Output |
---------------------------------------------------------------
- T
- A real value to be mapped onto the
interpolating curve. Values of T between zero
and one interpolate the original data; any
values of T outside this range result in extrapolation.
- N
- The number of points which were specified to determine
the curve in CURVS1DP.
- PARAM
- The arc lengths as computed by CURVS1DP.
- X
- An array containing the X-coordinates for the input
data as input to CURVS1DP. These need not be increasing.
- Y
- An array containing the Y-coordinates for the input data as
input to CURVS1DP.
- XP
- An array of second derivative values as calculated by CURVS1DP.
- YP
- An array of second derivative values as calculated by CURVS1DP.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- XO
- The interpolated X value.
- YO
- The interpolated Y value.
CURVPSDP - calculate values for a smoothing spline
(Double precision version of CURVPS)
This subroutine calculates certain values that are used by CURVP2DP in order to compute an interpolatory
smoothing spline under tension through a sequence of functional values
for a periodic function. The actual computation of the interpolated
values must be done using CURVP2DP.
Three parameters are used to control the degree of smoothness -- D, S,
and EPS.
The parameter D is a value indicating the degree of confidence in the
accuracy of the input function values -- it should be an approximation
of the standard deviation of error. Effectively the value of D
controls how close the smoothed curve comes to the input data points.
If D is small then the interpolated curve will pass close to the input
data. The larger the value of D, the more freedom the smooth curve
has in how close it comes to the input data values.
S is a more subtle global smoothing parameter. S must be
non-negative. For small values of S, the curve approximates the
tension spline and for larger values of S, the curve is smoother. A
reasonable value for S is REAL(N).
EPS controls the precision to which S is interpreted; EPS must be
between 0. and 1. inclusive. A reasonable value for EPS is
SQRT(2./REAL(N)).
---------------------------------------------------------------
Argument | Type | Mode | Dimension
---------------------------------------------------------------
CALL CURVPSDP (N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
P, | Double precision | Input |
D, | Double precision | Input | N
ISW, | Integer | Input |
S, | Double precision | Input |
EPS, | Double precision | Input |
YS, | Double precision | Output | N
YSP, | Double precision | Output | N
SIGMA, | Double precision | Input |
TEMP, | Double precision | Input | N x 11
IER) | Integer | Output |
---------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the abscissae for the input function.
- Y
- An array containing the functional values of the input function --
Y(K) is the functional value at X(K) for K=1,N.
- P
- The period of the function. P must be greater than
X(N)-X(1).
- D
- A user-specified value containing the observed weights. D may
either be an array or a scalar, depending on the value of ISW.
- ISW
- A switch for interpreting the value of D. If ISW=0, then D is
an array of length N (an error estimate for each input data value);
if ISW=1, then D is a scalar that serves as an error estimate
for every single data item.
- S
- Contains the value for smoothing. S must be non-negative. Larger
values for S yield greater smoothing. A reasonable value is REAL(N).
- EPS
- Contains a tolerance value for the relative precision to which
S should be interpreted. EPS must be between 0. and 1. inclusive.
A reasonable value is SQRT(2./REAL(N)).
- YS
- An array of length N.
- YSP
- An array of length N.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- TEMP
- Scratch space.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if S is negative.
- = 3 if EPS is negative or greater than 1.
- = 4 if X values are not strictly increasing.
- = 5 if D is negative.
- = 6 if P is less than or equal to X(N)-X(1).
KURV1DP - calculate values for KURV2 usage
(Double precision version of KURV1)
This subroutine calculates certain values that are used by KURV2DP in order to compute an interpolatory
spline under tension through a sequence of points in the plane. The
actual computation of the interpolated values must be done using KURV2DP.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURV1DP (N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
SLP1, | Double precision | Input |
SLPN, | Double precision | Input |
ISLPSW, | Integer | Input |
XP, | Double precision | Output | N
YP, | Double precision | Output | N
TEMP, | Double precision | Input | N
S, | Double precision | Output | N
SIGMA, | Double precision | Input |
IER) | Integer | Output |
--------------------------------------------------------------
- N
- The number of input data values. (N > 1)
- X
- An array containing the X values of the points.
- Y
- An array containing the Y values of the points.
Adjacent pairs of points must be distinct.
- SLP1
- A value (in radians) containing the slope at (X(1),Y(1)). The
angles are measured counter-clockwise from the X axis and the
positive sense of the curve is assumed to be that moving from
point 1 to point N. A value for SLP1 may be omitted as indicated
by the switch ISLPSW.
- SLPN
- A value (in radians) containing the slope at (X(N),Y(N)). The
angles are measured counter-clockwise from the X axis and the
positive sense of the curve is assumed to be that moving from
point 1 to point N. A value for SLP2 may be omitted as indicated
by the switch ISLPSW.
- ISLPSW
- A switch to indicate whether the slopes at the end points should
be calculated internally.
- = 0 if SLP1 and SLPN are user-specified.
- = 1 if SLP1 is user-specified, but SLPN calculated internally.
- = 2 if SLPN is user-specified, but SLP1 calculated internally.
- = 3 if SLP1 and SLPN are internally calculated.
- XP
- An array of length N.
- YP
- An array of length N.
- TEMP
- Scratch space.
- S
- An array of length N. On output, S contains the polygonal arclengths
of the curve.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if a pair of adjacent points coincide.
KURV2DP - does interpolation for parametric curves
(Double precision version of KURV2)
The interpolated curve is parameterized by mapping points in the
interval [0.,1.] onto the interpolated curve. The resulting curve has
a parametric representation both of whose components are splines under
tension and functions of the polygonal arc length. The value 0. is
mapped onto (X[1],Y[1]) and the value 1. is mapped onto (X[N],Y[N]).
KURV1DP must be called before calling KURV2DP.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURV2DP (T, | Double precision | Input |
XS, | Double precision | Output |
YS, | Double precision | Output |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
XP, | Double precision | Input | N
YP, | Double precision | Input | N
S, | Double precision | Input | N
SIGMA) | Double precision | Input |
--------------------------------------------------------------
- T
- A value to be mapped to a point on the curve. The interval
[0.,1.] is mapped onto the curve such that 0. is mapped to
(X(1),Y(1)) and 1. is mapped to (X(N),Y(N)). Values outside
of [0.,1.] result in extrapolation.
- XS
- Contains the X coordinate of the interpolated point that T maps to.
- YS
- Contains the Y coordinate of the interpolated point that T maps to.
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the input points.
- Y
- An array containing the Y values of the input points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N as computed by KURV1DP.
- YP
- An array of length N as computed by KURV1DP.
- S
- An array of length N containing the polygonal arc lengths of the
curve.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
KURVDDP - interpolation and derivatives for parametric curves
(Double precision version of KURVD)
This subroutine behaves like KURV2DP except
that in addition it returns the first and second derivatives of the
component functions in the parameterization. The interval [0.,1.]
maps onto a curve in the plane. The resulting curve has a parametric
representation both of whose components are splines under tension and
functions of the polygonal arc length. KURV1DP
must be called before calling KURVDDP.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL KURVDDP (T, | Double precision | Input |
XS, | Double precision | Output |
YS, | Double precision | Output |
XST, | Double precision | Output |
YST, | Double precision | Output |
XSTT, | Double precision | Output |
YSTT, | Double precision | Output |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
XP, | Double precision | Input | N
YP, | Double precision | Input | N
S, | Double precision | Input | N
SIGMA) | Double precision | Input |
--------------------------------------------------------------
- T
- A value to be mapped to a point on the curve. The interval
[0.,1.] is mapped onto the curve such that 0. is mapped to
(X(1),Y(1)) and 1. is mapped to (X(N),Y(N)). Values outside
of [0.,1.] result in extrapolation.
- XS
- Contains the X coordinate of the point that T maps to.
- YS
- Contains the Y coordinate of the point that T maps to.
- XST
- Contains the first derivatives of the X component with
respect to T.
- YST
- Contains the first derivatives of the Y component with
respect to T.
- XSTT
- Contains the second derivatives of the X component with
respect to T.
- YSTT
- Contains the second derivatives of the X component with
respect to T.
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the input points.
- Y
- An array containing the Y values of the input points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N as computed by KURV1DP.
- YP
- An array of length N as computed by KURV1DP.
- S
- An array of length N containing the polygonal arc lengths of the
curve, as computed by KURV1DP.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
KURVP1DP - calculate values for KURVP2 usage
(Double precision version of KURVP1)
This subroutine calculates certain values that are used by KURVP2DP in order to compute an interpolatory
spline under tension through a sequence of points in the plane that
form a closed curve. The actual computation of the interpolated
values must be done using KURVP2DP.
---------------------------------------------------------------
Argument | Type | Mode | Dimension
---------------------------------------------------------------
CALL KURVP1DP (N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
XP, | Double precision | Output | N
YP, | Double precision | Output | N
TEMP, | Double precision | Input | N x 2
S, | Double precision | Output | N
SIGMA, | Double precision | Input |
IER) | Integer | Output |
---------------------------------------------------------------
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the points.
- Y
- An array containing the Y values of the points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N.
- YP
- An array of length N.
- TEMP
- Scratch space.
- S
- An array of length N.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if a pair of adjacent points coincide.
KURVP2DP - does interpolation for closed
parametric curves
(Double precision version of KURVP2)
The interpolated curve is parameterized by mapping points in the
interval [0.,1.] onto an interpolated closed curve. The resulting
curve has a parametric representation both of whose components are
splines under tension and functions of the polygonal arc length. The
value 0. is mapped onto (X[1],Y[1]) and the value 1. is also mapped
onto (X[1],Y[1]), since the interpolated curve is closed. KURVP1DP must be called before calling KURVP2DP.
---------------------------------------------------------------
Argument | Type | Mode | Dimension
---------------------------------------------------------------
CALL KURVP2DP (T, | Double precision | Input |
XS, | Double precision | Output |
YS, | Double precision | Output |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
XP, | Double precision | Input | N
YP, | Double precision | Input | N
S, | Double precision | Input | N
SIGMA) | Double precision | Input |
---------------------------------------------------------------
- T
- A value to be mapped to a point on the curve. The interval
[0.,1.] is mapped onto the curve such that 0. is mapped to
(X(1),Y(1)) and 1. is mapped to (X(1),Y(1)), since the
interpolated curve is closed. Any interval [TT,TT+1.]
maps onto the entire curve.
- XS
- Contains the X coordinate of the point that T maps to.
- YS
- Contains the Y coordinate of the point that T maps to.
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the points.
- Y
- An array containing the Y values of the points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N as computed by KURV1DP.
- YP
- An array of length N as computed by KURV1DP.
- S
- An array of length N containing the polygonal arc lengths of the
curve, as computed by KURV1DP.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
KURVPDDP - interpolation and derivatives for closed parametric curves
(Double precision version of KURVPD)
This subroutine behaves like KURVP2DP except
that in addition it returns the first and second derivatives of the
component functions in the parameterization. The interval [0.,1.]
maps onto a closed curve in the plane. The resulting curve has a
parametric representation both of whose components are splines under
tension and functions of the polygonal arc length. KURVP1DP must be called before calling KURVPDDP.
---------------------------------------------------------------
Argument | Type | Mode | Dimension
---------------------------------------------------------------
CALL KURVPDDP (T, | Double precision | Input |
XS, | Double precision | Output |
YS, | Double precision | Output |
XST, | Double precision | Output |
YST, | Double precision | Output |
XSTT, | Double precision | Output |
YSTT, | Double precision | Output |
N, | Integer | Input |
X, | Double precision | Input | N
Y, | Double precision | Input | N
XP, | Double precision | Input | N
YP, | Double precision | Input | N
S, | Double precision | Input | N
SIGMA) | Double precision | Input |
---------------------------------------------------------------
- T
- A value to be mapped to a point on the curve. The interval
[0.,1.] is mapped onto the curve such that 0. is mapped to
(X(1),Y(1)) and 1. is mapped to (X(1),Y(1)). Any interval
[TT,TT+1.] maps onto the entire closed curve.
- XS
- Contains the X coordinate of the point that T maps to.
- YS
- Contains the Y coordinate of the point that T maps to.
- XST
- Contains the first derivatives of the X component with
respect to T.
- YST
- Contains the first derivatives of the Y component with
respect to T.
- XSTT
- Contains the second derivatives of the X component with
respect to T.
- YSTT
- Contains the second derivatives of the X component with
respect to T.
- N
- The number of input data points. (N > 1)
- X
- An array containing the X values of the input points.
- Y
- An array containing the Y values of the input points.
Adjacent pairs of points must be distinct.
- XP
- An array of length N as computed by KURVP1DP.
- YP
- An array of length N as computed by KURVP1DP.
- S
- An array of length N containing the polygonal arc lengths of the
curve, as computed by KURVP1DP.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
SURF1DP - calculate values for SURF2DP usage
(Double precision version of SURF1)
This subroutine calculates certain values that are used by
SURF2DP
in order to compute an interpolatory surface passing through a
rectangular grid of function values. The surface computed is a
tensor product of splines under tension. To calculate actual
interpolated values, SURF2DP must be called.
--------------------------------------------------------------
Argument | Type | Mode | Dimension
--------------------------------------------------------------
CALL SURF1DP (M, | Integer | Input |
N, | Integer | Input |
X, | Double precision | Input | M
Y, | Double precision | Input | N
Z, | Double precision | Input | IZ x N
IZ, | Integer | Input |
ZX1, | Double precision | Input | N
ZXM, | Double precision | Input | N
ZY1, | Double precision | Input | M
ZYN, | Double precision | Input | M
ZXY11, | Double precision | Input |
ZXYM1, | Double precision | Input |
ZXY1N, | Double precision | Input |
ZXYMN, | Double precision | Input |
ISLPSW, | Integer | Input |
ZP, | Double precision | Output | M x N x 3
TEMP, | Double precision | Input | N + N + M
SIGMA, | Double precision | Input |
IER) | Integer | Output |
--------------------------------------------------------------
- M
- The number of grid lines in the X direction. (M > 1)
- N
- The number of grid lines in the Y direction. (N > 1)
- X
- An array containing M X coordinates for grid lines in the X direction.
These values must be strictly increasing.
- Y
- An array containing N Y coordinates for grid lines in the Y direction.
These values must be strictly increasing.
- Z
- An array containing M x N functional values at the grid points;
Z(I,J) contains the functional value at (X(I),Y(J)) for I=1,M and
J=1,N.
- IZ
- The row dimension of the matrix Z (IZ is greater than or equal to M).
- ZX1
- An array containing N X-partial derivatives of the function along
the line X(1), that is ZX1(J) is the X-partial derivative at point
(X(1),Y(J)) for J=1,N. This parameter may be defaulted by setting
ISLPSW appropriately.
- ZXM
- An array containing N X-partial derivatives of the function along
the line X(M), that is ZXM(J) is the X-partial derivative at point
(X(M),Y(J)) for J=1,N. This parameter may be defaulted by setting
ISLPSW appropriately.
- ZY1
- An array containing M Y-partial derivatives of the function along
the line Y(1), that is ZY1(I) is the Y-partial derivative at point
(X(I),Y(1)) for I=1,M. This parameter may be defaulted by setting
ISLPSW appropriately.
- ZYN
- An array containing M Y-partial derivatives of the function along
the line Y(N), that is ZY1(I) is the Y-partial derivative at point
(X(I),Y(N)) for I=1,M. This parameter may be defaulted by setting
ISLPSW appropriately.
- ZXY11
- The X-Y-partial derivative at (X(1),Y(1). This parameter may
be defaulted by setting ISLPSW appropriately.
- ZXYM1
- The X-Y-partial derivative at (X(M),Y(1). This parameter may
be defaulted by setting ISLPSW appropriately.
- ZXY1N
- The X-Y-partial derivative at (X(1),Y(N). This parameter may
be defaulted by setting ISLPSW appropriately.
- ZXYMN
- The X-Y-partial derivative at (X(M),Y(N). This parameter may
be defaulted by setting ISLPSW appropriately.
- ISLPSW
- A switch to indicate which boundary derivatives are user
supplied and which should be estimated internally. Where
- I1 = 0 if ZX1 is user-supplied and 1 otherwise.
- I2 = 0 if ZXM is user-supplied and 1 otherwise.
- I3 = 0 if ZY1 is user-supplied and 1 otherwise.
- I4 = 0 if ZYN is user-supplied and 1 otherwise.
- I5 = 0 if ZXY11 is user-supplied and 1 otherwise.
- I6 = 0 if ZXYM1 is user-supplied and 1 otherwise.
- I7 = 0 if ZXY1N is user-supplied and 1 otherwise.
- I8 = 0 if ZXYMN is user-supplied and 1 otherwise.
set ISLPSW = I1 + 2*I2 + 4*I3 +8*I4 +16*I5 +32*I6 + 64*I7 + 128*I8.
Then, for example, if ISLPSW=0, then all derivative information
is user-supplied and if
ISLPSW=255, then all derivative information is to be internally
estimated.
- ZP
- An array of size M x N x 3. On output this contains partial
derivatives of the surface at the given nodes. ZP is used by
SURF2DP.
- TEMP
- Scratch space.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
- IER
- An error return value. If IER is returned as 0, then no
errors were detected.
- = 1 if N is less than 2.
- = 2 if X or Y values are not strictly increasing.
SURF2DP - 2D interpolation for gridded data
(Double precision version of SURF2)
This function interpolates a surface value at a specified coordinate
using bi-splines under tension. SURF1DP must
be called before invoking SURF2DP. The desired interpolated value is
returned as the value of the function.
------------------------------------------------------------------
Argument | Type | Mode | Dimension
------------------------------------------------------------------
FUNCTION SURF2DP (XX, | Double precision | Input |
YY, | Double precision | Input |
M, | Integer | Input |
N, | Integer | Input |
X, | Double precision | Input | M
Y, | Double precision | Input | N
Z, | Double precision | Input | IZ x N
IZ, | Integer | Input |
ZP, | Double precision | Input | M x N x 3
SIGMA) | Double precision | Input |
------------------------------------------------------------------
- XX
- Contains the X coordinate of a point to be mapped onto the interpolated
surface.
- YY
- Contains the Y coordinate of a point to be mapped onto the interpolated
surface.
- M
- The number of grid lines in the X direction. (M > 1)
- N
- The number of grid lines in the Y direction. (N > 1)
- X
- An array containing M X coordinates for grid lines in the X direction.
These values must be strictly increasing.
- Y
- An array containing N Y coordinates for grid lines in the Y direction.
These values must be strictly increasing.
- Z
- An array containing M x N functional values at the grid points;
Z(I,J) contains the functional value at (X(I),Y(J)) for I=1,M and
J=1,N.
- IZ
- The row dimension of the matrix Z (IZ is greater than or equal to M).
- ZP
- An array of size M x N x 3.
- SIGMA
- Tension factor. Values near zero result in a cubic spline;
large values (e.g. 30) result in nearly a polygonal line. A
typical value is 1.
Values of SIGMA larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
c_ftcurv - 1D interpolation for non-periodic
functions
This function calculates an interpolatory spline under tension through
a sequence of functional values.
c_ftcurv is called after all of the desired values for
control parameters have been set using the procedures
c_ftseti,
c_ftsetr,
c_ftsetrd,
c_ftsetc.
Control parameters that apply to c_ftcurv are:
sig,
sl1,
sln,
sf1.
The value for the parameter sig specifies the tension factor. Values near
zero result in a cubic spline; large values (e.g. 30)
result in nearly a polygonal line. A typical value is 1. (the default).
Values of sig larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
The values for sl1 and sln specify the slope of the curve at the first
point and last point, respectively.
The value of sf1 controls whether to use the values for sl1 and sln, or
compute those values internally. Specifically, sf1
- = 0 if sl1 and sln are user-specified.
- = 1 if sl1 is user-specified, but sln is internally calculated.
- = 2 if sln is user-specified, but sl1 is internally calculated.
- = 3 if sl1 and sln are internally calculated.
By default the slopes at the end points are computed internally.
Function prototype:
int c_ftcurv (int, float [], float [], int, float [], float []);
Return value:
c_ftcurv returns an error value as per:
- = 0 -- no error.
- = 1 -- if n is less than 2.
- = 2 -- if X values are not strictly increasing.
Argument description:
-------------------------------------------------
Argument | Type | Size
-------------------------------------------------
int c_ftcurv (n, | int |
xi, | float [] | n
yi, | float [] | n
m, | int |
xo, | float [] | m
yo | float [] | m
);
-------------------------------------------------
- n
- The number of input data points. (n > 1)
- xi
- An array containing the abscissae for the input function.
- yi
- An array containing the functional values of the input function --
yi[k] is the functional value at xi[k] for k=0,n-1.
- m
- The number of desired interpolated values.
- xo
- An array of length m containing the abscissae for the interpolated values.
- yo
- An array containing the interpolated functional values -- yo[k] is
the functional value at xo[k] for k=0,m-1.
c_ftcurvd - calculate derivatives
This function calculates the derivatives of an interpolatory spline
under tension.
c_ftcurvd is called after all of the desired values for
control parameters have been set using the procedures
c_ftseti,
c_ftsetr,
c_ftsetrd,
c_ftsetc.
Control parameters that apply to c_ftcurvd are:
sig,
sl1,
sln,
sf1.
The value for the parameter sig specifies the tension factor. Values near
zero result in a cubic spline; large values (e.g. 30)
result in nearly a polygonal line. A typical value is 1. (the default).
Values of sig larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
The values for sl1 and sln specify the slope of the curve at the first
point and last point, respectively.
The value of sf1 controls whether to use the values for sl1 and sln, or
compute those values internally. Specifically, sf1
- = 0 if sl1 and sln are user-specified.
- = 1 if sl1 is user-specified, but sln is internally calculated.
- = 2 if sln is user-specified, but sl1 is internally calculated.
- = 3 if sl1 and sln are internally calculated.
By default the slopes at the end points are computed internally.
Function prototype:
int c_ftcurvd (int, float [], float [], int, float [], float []);
Return value:
c_ftcurvd returns an error value as per:
- = 0 -- no error.
- = 1 -- if N is less than 2.
- = 2 -- if X values are not strictly increasing.
Argument description:
-------------------------------------------------
Argument | Type | Size
-------------------------------------------------
int c_ftcurvd (n, | int |
xi, | float [] | n
yi, | float [] | n
m, | int |
xo, | float [] | m
yo | float [] | m
);
-------------------------------------------------
- n
- The number of input data points. (n > 1)
- xi
- An array containing the abscissae for the input function.
- yi
- An array containing the functional values (y[k] is the functional
value at x[k] for k=0,n).
- m
- The number of desired derivatives.
- xo
- An array containing the abscissae for the output points.
- yo
- An array containing the interpolated derivative values -- yo[k] is
the functional derivative at xo[k] for k=0,n.
c_ftcurvi - calculate integrals
This function calculates integrals of an interpolatory spline
under tension between two user-specified limits.
c_ftcurvi is called after all of the desired values for
control parameters have
been set using the procedures
c_ftseti,
c_ftsetr,
c_ftsetrd,
c_ftsetc.
Control parameters that apply to c_ftcurvd are:
sig,
sl1,
sln,
sf1.
The value for the parameter sig specifies the tension factor. Values near
zero result in a cubic spline; large values (e.g. 30)
result in nearly a polygonal line. A typical value is 1. (the default).
Values of sig larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
The values for sl1 and sln specify the slope of the curve at the first
point and last point, respectively.
The value of sf1 controls whether to use the values for sl1 and sln, or
compute those values internally. Specifically, sf1
- = 0 if sl1 and sln are user-specified.
- = 1 if sl1 is user-specified, but sln is internally calculated.
- = 2 if sln is user-specified, but sl1 is internally calculated.
- = 3 if sl1 and sln are internally calculated.
By default the slopes at the end points are computed internally.
Function prototype:
int c_ftcurvi (float, float, int, float [], float [], float *);
Return value:
c_ftcurvi returns an error value as per:
- = 0 -- no error.
- = 1 -- if n is less than 2.
- = 2 -- if X values are not strictly increasing.
Argument description:
-------------------------------------------------
Argument | Type | Size
-------------------------------------------------
int c_ftcurvi (xl, | float |
xr, | float |
n, | int |
xi, | float [] | n
yi, | float [] | n
integral | float * |
);
-------------------------------------------------
- xl
- The lower limit of the integration.
- xr
- The upper limit of the integration.
- n
- The number of input data points. (N > 1)
- xi
- An array containing the abscissae for the input function.
- yi
- An array containing the functional values of the input function
(y[k] is the functional value at x[k] for k=0,n).
- integral
- The integral of the function from xl to xr is given by *integral.
c_ftcurvp - 1D interpolation for periodic functions
This function calculates an interpolatory spline under tension through
a sequence of functional values for a periodic function.
c_ftcurvp is called after all of the desired values for
control parameters have been set using the procedures
c_ftseti,
c_ftsetr,
c_ftsetrd,
c_ftsetc.
The only control parameter that applies to c_ftcurvp is:
sig.
The value for the parameter sig specifies the tension factor. Values near
zero result in a cubic spline; large values (e.g. 30)
result in nearly a polygonal line. A typical value is 1. (the default).
Values of sig larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
Function prototype:
int c_ftcurvp (int, float [], float [], float, int, float [], float []);
Return value:
c_ftcurvp returns an error value as per:
- = 0 -- no error.
- = 1 -- if n is less than 2.
- = 2 -- if the period is strictly less than the span of the abscissae.
Argument description:
-------------------------------------------------
Argument | Type | Size
-------------------------------------------------
int c_ftcurvp (n, | int |
xi, | float [] | n
yi, | float [] | n
p, | float |
m, | int |
xo, | float [] | m
yo | float [] | m
);
-------------------------------------------------
- n
- The number of input data points. (n > 1)
- xi
- An array containing the abscissae for the input function.
- yi
- An array containing the input functional values (y[k] is the functional
value at x[k] for k=0,n).
- p
- The period of the function; p must not
be less than xi[n-1] - xi[0].
- m
- The number of desired interpolated points.
- xo
- An array containing the abscissae for the interpolated values.
- yo
- An array containing the interpolated functional values (yo[k] is
the functional value at xo[k] for k=0,n).
c_ftcurvpi - calculate integrals for periodic
functions
This function calculates an integral between two specified points.
c_ftcurvpi is called after all of the desired values for
control parameters have been set using the procedures
c_ftseti,
c_ftsetr,
c_ftsetrd,
c_ftsetc.
The only control parameter that applies to c_ftcurvpi is:
sig.
The value for the parameter sig specifies the tension factor. Values near
zero result in a cubic spline; large values (e.g. 30)
result in nearly a polygonal line. A typical value is 1. (the default).
Values of sig larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
Function prototype:
int c_ftcurvpi (float, float, float, int, float [], float [], float *);
Return value:
c_ftcurpi returns an error value as per:
- = 0 -- no error.
- = 1 -- if n is less than 2.
- = 2 -- if the period is strictly less than the span of the abscissae.
Argument description:
-------------------------------------------------
Argument | Type | Size
-------------------------------------------------
int c_ftcurvpi (xl, | float |
xr, | float |
p, | float |
m, | int |
xi, | float [] | m
yi, | float [] | m
integral | float * |
);
-------------------------------------------------
- xl
- The lower limit of the integration.
- xr
- The upper limit of the integration.
- p
- The period of the function; p must not be less than xi[n-1] - xi[0].
- m
- The number of input points.
- xi
- An array containing the abscissae for the input points.
- yi
- An array containing the values of the input points (yi[k] is
the functional value at xi[k] for k=0,m-1).
- integral
- The desired integral value is returned in *integral.
c_ftcurvs - smoothing spline
This function
computes an interpolatory smoothing spline under tension through
a sequence of functional values.
Two parameters and one function argument used to
control the degree of smoothness -- the parameters are smt, and eps and the
function argument is d.
The argument d is a value indicating the degree of confidence
in the accuracy of the input function values -- it should be an
approximation of the standard deviation of error. Effectively the value
of d controls how close the smoothed curve comes to the input data
points. If d is small, then the interpolated curve
will pass close to the
input data. The larger the value of d, the more freedom the smooth
curve has in how close it comes to the input data values.
The parameter smt is a
more subtle global smoothing parameter; smt must be non-negative.
For small
values of smt, the curve approximates the tension spline and for larger
values of smt, the
curve is smoother. A reasonable value for smt is (float) n.
The parameter eps controls the precision to which smt is
interpreted; eps
must be between 0. and 1. inclusive. A reasonable value for eps is
sqrt( 2./(float) n ).
c_ftcurvs is called after all of the desired values for
control parameters have been set using the procedures
c_ftseti,
c_ftsetr,
c_ftsetrd,
c_ftsetc.
Control parameters that apply to c_ftcurvs are:
sig,
smt,
eps,
sf2.
The value for the parameter sig specifies the tension factor. Values near
zero result in a cubic spline; large values (e.g. 30)
result in nearly a polygonal line. A typical value is 1. (the default).
Values of sig larger than about 40. can result in an
internal floating point overflow on 32-bit machines, so
you should keep SIGMA below that number on such machines.
Function prototype:
int c_ftcurvs (int, float [], float [], int, float [], int,
float [], float []);
Return value:
c_ftcurvs returns an error value as per:
- = 1 if N is less than 2.
- = 2 if smt is negative.
- = 3 if eps is negative or greater than 1.
- = 4 if X values are not strictly increasing.
- = 5 if d is negative.
Argument description:
--------------------------------------------------------------
Argument | Type | Size
--------------------------------------------------------------
int c_ftcurvs (n, | int |
xi, | float [] | n
yi, | float [] | n
dflg, | int |
d, | float [] | n
m, | int |
xo, | float [] | m
yo | float [] | m
);
--------------------------------------------------------------
- n
- The number of input data points. (n > 1)
- xi
- An array containing the abscissae for the input function.
- yi
- An array containing the functional values of the input
function (yi[k] is the functional value at x[k] for k=0,n-1).
- dflg
- A switch for interpreting the value of d (as described below).
If dflg=0, then d is
an array of length n (an error estimate for each input data value);
if dflg=1, then d is a scalar that serves as an error estimate
for every single data item.
- d
- A user-specified value containing the observed weights. d may
either be an array or a scalar, depending on the value of dflg.
- m
- The number of output values.
- xo
- Contains the abscissae for the output values.
- yo
- Contains the functional values for the smoothing spline
((yo[k] is the functional value at xo[k] for k=0,n-1).
c_ftcurvs1 - calculate values for a smoothing spline for data in the plane.
This function
computes an interpolatory smoothing spline under tension through
a sequence of values in the plane.
Two parameters and one function argument are used to
control the degree of smoothness -- the parameters are smt, and eps and the
function argument is d.
The argument d is a value indicating the degree of confidence
in the accuracy of the input function values -- it should be an
approximation of the standard deviation of error. Effectively the value
of d controls how close the smoothed curve comes to the input data
points. If d is small, then the interpolated curve
will pass close to the
input data. The larger the value of d, the more freedom the smooth
curve has in how close it comes to the input data values.
The parameter smt is a
more subtle global smoothing parameter; smt must be non-negative.
For small
values of smt, the curve approximates the tension spline and for larger
values of smt, the
curve is smoother.