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

Function Return Value

QDVAL — Value of the quadratic interpolant at X.   (Output)

Required Arguments

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 3.
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:          QDVAL (X, XDATA, FDATA [,…])

Specific:         The specific interface names are S_QDVAL and D_QDVAL.

FORTRAN 77 Interface

Single:            QDVAL (X, NDATA, XDATA, FDATA, CHECK)

Double:          The double precision name is DQDVAL.

Description

The function QDVAL interpolates a table of values, using quadratic polynomials, returning an approximation to 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 QDVAL is q(x).

Comments

Informational error

Type   Code

4           3                  The XDATA values must be strictly increasing.

Example

In this example, the value of sin x is approximated at π/4 by using QDVAL on a table of 33 equally spaced values.

 

      USE IMSL_LIBRARIES

 

      IMPLICIT   NONE

      INTEGER    NDATA

      PARAMETER  (NDATA=33)

!

      INTEGER    I, NOUT

      REAL       F, FDATA(NDATA), H, PI, QT, SIN, X,&

                 XDATA(NDATA)

      INTRINSIC  SIN

!                                  Define function

      F(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

!                                  Evaluate at PI/4

      QT = QDVAL(X,XDATA,FDATA)

!                                  Get output unit number

      CALL UMACH (2, NOUT)

!                                  Print results

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

!

99999 FORMAT (15X, 'X', 6X, 'F(X)', 6X, 'QDVAL', 5X, 'ERROR', //, 6X,&

             4F10.3, /)

      END

Output

 

    X      F(X)      QDVAL     ERROR

0.785     0.707     0.707     0.000

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