This function evaluates the derivative of a function defined on a set of points using quadratic interpolation.

Function Return Value

QDDER — Value of the IDERIV-th derivative of the quadratic interpolant at X.   (Output)

Required Arguments

IDERIV — Order of the derivative.   (Input)

X — Coordinate of the point at which the function is to be evaluated.   (Input)

XDATA — Array of length NDATA containing the location of the data points.   (Input) XDATA must be strictly increasing.

FDATA — Array of length NDATA containing the function values.   (Input)
FDATA(I) is the value of the function at XDATA(I).

Optional Arguments

NDATA — Number of data points.   (Input)
NDATA must be at least three.
Default: NDATA = size (XDATA,1).

CHECK — Logical variable that is .TRUE. if checking of XDATA is required or .FALSE. if checking is not required.   (Input)
Default: CHECK = .TRUE.

FORTRAN 90 Interface

Generic:          QDDER(IDERIV, X, XDATA, FDATA [,…])

Specific:         The specific interface names are S_QDVAL and D_QDVAL.

FORTRAN 77 Interface

Single:            QDDER(IDERIV, X, NDATA, XDATA, FDATA, CHECK)

Double:          The double precision function name is DQDVAL.

Description

The function QDDER interpolates a table of values, using quadratic polynomials, returning an approximation to the derivative of the tabulated function. Let (xi, fi) for i = 1, …, n be the tabular data. Given a number x at which an interpolated value is desired, we first find the nearest interior grid point xi. A quadratic interpolant q is then formed using the three points (xi-1, fi-1)
(xi, fi), and (xi+1, fi+1). The number returned by QDDER is q(j)(x), where j = IDERIV.

Comments

1.         Informational error

Type   Code

4           3                  The XDATA values must be strictly increasing.

2.         Because quadratic interpolation is used, if the order of the derivative is greater than two, then the returned value is zero.

Example

In this example, the value of sin x and its derivatives are approximated at π/4 by using QDDER on a table of 33 equally spaced values.

 

      USE IMSL_LIBRARIES

 

      IMPLICIT   NONE

      INTEGER    NDATA

      PARAMETER  (NDATA=33)

!

      INTEGER    I, IDERIV, NOUT

      REAL       COS, F, F1, F2, FDATA(NDATA), H, PI,&

                 QT, SIN, X, XDATA(NDATA)

      LOGICAL CHECK   

      INTRINSIC  COS, SIN

!                                  Define function and derivatives

      F(X)  = SIN(X)

      F1(X) = COS(X)

      F2(X) = -SIN(X)

!                                  Generate data points

      XDATA(1) = 0.0

      FDATA(1) = F(XDATA(1))

      H        = 1.0/32.0

      DO 10  I=2, NDATA

         XDATA(I) = XDATA(I-1) + H

         FDATA(I) = F(XDATA(I))

   10 CONTINUE

!                                  Get value of PI and set X

      PI = CONST('PI')

      X  = PI/4.0

!                                  Check XDATA

      CHECK = .TRUE.

!                                  Get output unit number

      CALL UMACH (2, NOUT)

!                                  Write heading

      WRITE (NOUT,99998)

!                                  Evaluate quadratic at PI/4

      IDERIV = 0

      QT     = QDDER(IDERIV,X,XDATA,FDATA, CHECK=CHECK)

      WRITE (NOUT,99999) X, IDERIV, F(X), QT, (F(X)-QT)

      CHECK = .FALSE.

!                                  Evaluate first derivative at PI/4

      IDERIV = 1

      QT     = QDDER(IDERIV,X,XDATA,FDATA)

      WRITE (NOUT,99999) X, IDERIV, F1(X), QT, (F1(X)-QT)

!                                  Evaluate second derivative at PI/4

      IDERIV = 2

      QT     = QDDER(IDERIV,X,XDATA,FDATA, CHECK=CHECK)

      WRITE (NOUT,99999) X, IDERIV, F2(X), QT, (F2(X)-QT)

!

99998 FORMAT (33X, 'IDER', /, 15X, 'X', 6X, 'IDER', 6X, 'F    (X)',&

             5X, 'QDDER', 6X, 'ERROR', //)

99999 FORMAT (7X, F10.3, I8, 3F12.3/)

      END

Output

 

                      IDER
   X      IDER      F    (X)     QDDER      ERROR

0.785       0       0.707       0.707       0.000

0.785       1       0.707       0.707       0.000

0.785       2      -0.707      -0.704      -0.003

Visual Numerics, Inc. Visual Numerics - Developers of IMSL and PV-WAVE http://www.vni.com/ PHONE: 713.784.3131 FAX:713.781.9260