This function evaluates the derivative of a three-dimensional tensor-product spline, given its tensor-product B-spline representation.
Function Return Value
BS3DR — Value of the (IXDER, IYDER, IZDER) derivative of the spline at (X, Y, Z). (Output)
Required Arguments
IXDER — Order of the X-derivative. (Input)
IYDER — Order of the Y-derivative. (Input)
IZDER — Order of the Z-derivative. (Input)
X — X-coordinate of the point at which the spline is to be evaluated. (Input)
Y — Y-coordinate of the point at which the spline is to be evaluated. (Input)
Z — Z-coordinate of the point at which the spline is to be evaluated. (Input)
KXORD — Order of the spline in the X-direction. (Input)
KYORD — Order of the spline in the Y-direction. (Input)
KZORD — Order of the spline in the Z-direction. (Input)
XKNOT — Array of
length NXCOEF +
KXORD containing
the knot sequence in the X-direction.
(Input)
KNOT
must be nondecreasing.
YKNOT — Array of
length NYCOEF +
KYORD containing
the knot sequence in the Y-direction.
(Input)
YKNOT must be
nondecreasing.
ZKNOT — Array of
length NZCOEF +
KZORD containing
the knot sequence in the Z-direction.
(Input)
ZKNOT must be
nondecreasing.
NXCOEF — Number of B-spline coefficients in the X-direction. (Input)
NYCOEF — Number of B-spline coefficients in the Y-direction. (Input)
NZCOEF — Number of B-spline coefficients in the Z-direction. (Input)
BSCOEF — Array of
length NXCOEF
* NYCOEF * NZCOEF containing the
tensor-product
B-spline coefficients. (Input)
BSCOEF is treated
internally as a matrix of size NXCOEF by NYCOEF by NZCOEF.
FORTRAN 90 Interface
Generic: BS3DR (IXDER, IYDER, IZDER, X, Y, Z, KXORD, KYORD, KZORD, XKNOT, YKNOT, ZKNOT, NXCOEF, NYCOEF, NZCOEF, BSCOEF)
Specific: The specific interface names are S_BS3DR and D_BS3DR.
FORTRAN 77 Interface
Single: BS3DR (IXDER, IYDER, IZDER, X, Y, Z, KXORD, KYORD, KZORD, XKNOT, YKNOT, ZKNOT, NXCOEF, NYCOEF, NZCOEF, BSCOEF)
Double: The double precision function name is DBS3DR.
Description
The function BS3DR evaluates a partial derivative of a trivariate tensor-product spline (represented as a linear combination of tensor-product B-splines) at a given point. For more information, see de Boor (1978, pages 351− 353).
This routine returns the value of the function s(p, q, r) at a point (x, y, z) given the coefficients c by computing
where kx, ky, and kz are the orders of the splines. (These numbers are passed to the subroutine in KXORD, KYORD, and KZORD, respectively.) Likewise, tx, ty, and tz are the corresponding knot sequences (XKNOT, YKNOT, and ZKNOT).
Comments
1. Workspace may be explicitly provided, if desired, by use of B23DR/DB23DR. The reference is:
CALL B23DR(IXDER, IYDER, IZDER, X, Y, Z, KXORD, KYORD, KZORD, XKNOT, YKNOT, ZKNOT, NXCOEF, NYCOEF, NZCOEF, BSCOEF, WK)
The additional argument is:
WK — Work array of length 3 * MAX0(KXORD, KYORD, KZORD) + KYORD * KZORD + KZORD.
2. Informational errors
Type Code
3
1
The point X does
not satisfy
XKNOT(KXORD) .LE. X .LE. XKNOT(NXCOEF + 1).
3
2
The point Y does
not satisfy
YKNOT(KYORD) .LE. Y .LE. YKNOT(NYCOEF + 1).
3
3
The point Z does
not satisfy
ZKNOT (KZORD) .LE. Z .LE. ZKNOT(NZCOEF + 1).
Example
In this example, a spline interpolant s to a function f(x, y, z) = x4 + y(xz)3 is constructed using BS3IN. Next, BS3DR is used to compute s(2,0,1)(x, y, z). The values of this partial derivative and the error are computed on a 4 × 4 × 2 grid and then displayed.
USE BS3DR_INT
USE BS3IN_INT
USE BSNAK_INT
USE UMACH_INT
IMPLICIT NONE
! SPECIFICATIONS FOR PARAMETERS
INTEGER KXORD, KYORD, KZORD, LDF, MDF, NXDATA, NXKNOT,&
NYDATA, NYKNOT, NZDATA, NZKNOT
PARAMETER (KXORD=5, KYORD=2, KZORD=3, NXDATA=21, NYDATA=6,&
NZDATA=8, LDF=NXDATA, MDF=NYDATA,&
NXKNOT=NXDATA+KXORD, NYKNOT=NYDATA+KYORD,&
NZKNOT=NZDATA+KZORD)
!
INTEGER I, J, K, L, NOUT, NXCOEF, NYCOEF, NZCOEF
REAL BSCOEF(NXDATA,NYDATA,NZDATA), F, F201,&
FDATA(LDF,MDF,NZDATA), FLOAT, S201, X, XDATA(NXDATA),&
XKNOT(NXKNOT), Y, YDATA(NYDATA), YKNOT(NYKNOT), Z,&
ZDATA(NZDATA), ZKNOT(NZKNOT)
INTRINSIC FLOAT
! Define function and (2,0,1)
! derivative
F(X,Y,Z) = X*X*X*X + X*X*X*Y*Z*Z*Z
F201(X,Y,Z) = 18.0*X*Y*Z
! Set up X-interpolation points
DO 10 I=1, NXDATA
XDATA(I) = FLOAT(I-11)/10.0
10 CONTINUE
! Set up Y-interpolation points
DO 20 I=1, NYDATA
YDATA(I) = FLOAT(I-1)/FLOAT(NYDATA-1)
20 CONTINUE
! Set up Z-interpolation points
DO 30 I=1, NZDATA
ZDATA(I) = FLOAT(I-1)/FLOAT(NZDATA-1)
30 CONTINUE
! Generate knots
CALL BSNAK (NXDATA, XDATA, KXORD, XKNOT)
CALL BSNAK (NYDATA, YDATA, KYORD, YKNOT)
CALL BSNAK (NZDATA, ZDATA, KZORD, ZKNOT)
! Generate FDATA
DO 50 K=1, NZDATA
DO 40 I=1, NYDATA
DO 40 J=1, NXDATA
FDATA(J,I,K) = F(XDATA(J),YDATA(I),ZDATA(K))
40 CONTINUE
50 CONTINUE
! Get output unit number
CALL UMACH (2, NOUT)
! Interpolate&
CALL BS3IN (XDATA, YDATA,
ZDATA, FDATA, KXORD, KYORD, KZORD, XKNOT,
&
YKNOT, ZKNOT, BSCOEF)
!
NXCOEF = NXDATA
NYCOEF = NYDATA
NZCOEF = NZDATA
! Write heading
WRITE (NOUT,99999)
! Print over a grid of
! [-1.0,1.0] x [0.0,1.0] x [0.0,1.0]
! at 32 points.
DO 80 I=1, 4
DO 70 J=1, 4
DO 60 L=1, 2
X = 2.0*(FLOAT(I-1)/3.0) - 1.0
Y = FLOAT(J-1)/3.0
Z = FLOAT(L-1)
! Evaluate spline
S201 = BS3DR(2,0,1,X,Y,Z,KXORD,KYORD,KZORD,XKNOT,YKNOT,&
ZKNOT,NXCOEF,NYCOEF,NZCOEF,BSCOEF)
WRITE (NOUT,'(3F12.4,2F12.6)') X, Y, Z, S201,&
F201(X,Y,Z) - S201
60 CONTINUE
70 CONTINUE
80 CONTINUE
99999 FORMAT (38X, '(2,0,1)', /, 9X, 'X', 11X,&
'Y', 11X, 'Z', 4X, 'S (X,Y,Z) Error')
END
Output
(2,0,1)
X
Y
Z S (X,Y,Z)
Error
-1.0000
0.0000 0.0000
-0.000107 0.000107
-1.0000
0.0000 1.0000
0.000053 -0.000053
-1.0000
0.3333 0.0000
0.064051 -0.064051
-1.0000
0.3333 1.0000 -5.935941
-0.064059
-1.0000
0.6667 0.0000
0.127542 -0.127542
-1.0000
0.6667 1.0000 -11.873034
-0.126966
-1.0000
1.0000 0.0000
0.191166 -0.191166
-1.0000
1.0000 1.0000 -17.808527
-0.191473
-0.3333
0.0000 0.0000
-0.000002 0.000002
-0.3333
0.0000 1.0000
0.000000 0.000000
-0.3333
0.3333 0.0000
0.021228 -0.021228
-0.3333
0.3333 1.0000 -1.978768
-0.021232
-0.3333
0.6667 0.0000
0.042464 -0.042464
-0.3333
0.6667 1.0000 -3.957536
-0.042464
-0.3333
1.0000 0.0000
0.063700 -0.063700
-0.3333
1.0000 1.0000 -5.936305
-0.063694
0.3333
0.0000
0.0000 -0.000003
0.000003
0.3333
0.0000 1.0000
0.000000
0.000000
0.3333
0.3333 0.0000
-0.021229
0.021229
0.3333
0.3333 1.0000
1.978763
0.021238
0.3333
0.6667 0.0000
-0.042465
0.042465
0.3333
0.6667 1.0000
3.957539
0.042462
0.3333
1.0000 0.0000
-0.063700
0.063700
0.3333
1.0000 1.0000
5.936304
0.063697
1.0000
0.0000 0.0000
-0.000098
0.000098
1.0000
0.0000 1.0000
0.000053 -0.000053
1.0000
0.3333 0.0000
-0.063855
0.063855
1.0000
0.3333 1.0000
5.936146
0.063854
1.0000
0.6667 0.0000
-0.127631
0.127631
1.0000
0.6667 1.0000
11.873067
0.126933
1.0000
1.0000 0.0000
-0.191442
0.191442
1.0000
1.0000 1.0000
17.807940 0.192060
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