## MC01ND

### Value of a real polynomial at a given complex point

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To compute the value of the real polynomial P(x) at a given
complex point x = x0 using Horner's algorithm.

```
Specification
```      SUBROUTINE MC01ND( DP, XR, XI, P, VR, VI, INFO )
C     .. Scalar Arguments ..
INTEGER           DP, INFO
DOUBLE PRECISION  VI, VR, XI, XR
C     .. Array Arguments ..
DOUBLE PRECISION  P(*)

```
Arguments

Input/Output Parameters

```  DP      (input) INTEGER
The degree of the polynomial P(x).  DP >= 0.

XR      (input) DOUBLE PRECISION
XI      (input) DOUBLE PRECISION
The real and imaginary parts, respectively, of x0.

P       (input) DOUBLE PRECISION array, dimension (DP+1)
This array must contain the coefficients of the polynomial
P(x) in increasing powers of x.

VR      (output) DOUBLE PRECISION
VI      (output) DOUBLE PRECISION
The real and imaginary parts, respectively, of P(x0).

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  Given the real polynomial
2                   DP
P(x) = p(1) + p(2) * x + p(3) * x + ... + p(DP+1) * x  ,

the routine computes the value of P(x0) using the recursion

q(DP+1) = p(DP+1),
q(i) = x0*q(i+1) + p(i) for i = DP, DP-1, ..., 1,

which is known as Horner's algorithm (see ). Then q(1) = P(x0).

```
References
```   STOER, J and BULIRSCH, R.
Introduction to Numerical Analysis.
Springer-Verlag. 1980.

```
Numerical Aspects
```  The algorithm requires DP operations for real arguments and 4*DP
for complex arguments.

```
```  None
```
Example

Program Text

```*     MC01ND EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          DPMAX
PARAMETER        ( DPMAX = 20 )
*     .. Local Scalars ..
DOUBLE PRECISION VI, VR, XI, XR
INTEGER          DP, I, INFO
*     .. Local Arrays ..
DOUBLE PRECISION P(DPMAX+1)
*     .. External Subroutines ..
EXTERNAL         MC01ND
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) DP, XR, XI
IF ( DP.LE.-1 .OR. DP.GT.DPMAX ) THEN
WRITE ( NOUT, FMT = 99995 ) DP
ELSE
READ ( NIN, FMT = * ) ( P(I), I = 1,DP+1 )
*        Evaluate the polynomial at the given (complex) point.
CALL MC01ND( DP, XR, XI, P, VR, VI, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) XR, XI, VR
WRITE ( NOUT, FMT = 99996 ) XR, XI, VI
END IF
END IF
*
STOP
*
99999 FORMAT (' MC01ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01ND = ',I2)
99997 FORMAT (' Real      part of P(',F6.2,SP,F6.2,'*j ) = ',SS,F8.4)
99996 FORMAT (/' Imaginary part of P(',F6.2,SP,F6.2,'*j ) = ',SS,F8.4)
99995 FORMAT (/' DP is out of range.',/' DP = ',I5)
END
```
Program Data
``` MC01ND EXAMPLE PROGRAM DATA
4     -1.56     0.29
5.0  3.0  -1.0  2.0  1.0
```
Program Results
``` MC01ND EXAMPLE PROGRAM RESULTS

Real      part of P( -1.56 +0.29*j ) =  -4.1337

Imaginary part of P( -1.56 +0.29*j ) =   1.7088
```