## MC01RD

### Polynomial operation P(x) = P1(x) P2(x) + alpha P3(x)

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

Purpose

```  To compute the coefficients of the polynomial

P(x) = P1(x) * P2(x) + alpha * P3(x),

where P1(x), P2(x) and P3(x) are given real polynomials and alpha
is a real scalar.

Each of the polynomials P1(x), P2(x) and P3(x) may be the zero
polynomial.

```
Specification
```      SUBROUTINE MC01RD( DP1, DP2, DP3, ALPHA, P1, P2, P3, INFO )
C     .. Scalar Arguments ..
INTEGER           DP1, DP2, DP3, INFO
DOUBLE PRECISION  ALPHA
C     .. Array Arguments ..
DOUBLE PRECISION  P1(*), P2(*), P3(*)

```
Arguments

Input/Output Parameters

```  DP1     (input) INTEGER
The degree of the polynomial P1(x).  DP1 >= -1.

DP2     (input) INTEGER
The degree of the polynomial P2(x).  DP2 >= -1.

DP3     (input/output) INTEGER
On entry, the degree of the polynomial P3(x).  DP3 >= -1.
On exit, the degree of the polynomial P(x).

ALPHA   (input) DOUBLE PRECISION
The scalar value alpha of the problem.

P1      (input) DOUBLE PRECISION array, dimension (lenp1)
where lenp1 = DP1 + 1 if DP1 >= 0 and 1 otherwise.
If DP1 >= 0, then this array must contain the
coefficients of P1(x) in increasing powers of x.
If DP1 = -1, then P1(x) is taken to be the zero
polynomial, P1 is not referenced and can be supplied
as a dummy array.

P2      (input) DOUBLE PRECISION array, dimension (lenp2)
where lenp2 = DP2 + 1 if DP2 >= 0 and 1 otherwise.
If DP2 >= 0, then this array must contain the
coefficients of P2(x) in increasing powers of x.
If DP2 = -1, then P2(x) is taken to be the zero
polynomial, P2 is not referenced and can be supplied
as a dummy array.

P3      (input/output) DOUBLE PRECISION array, dimension (lenp3)
where lenp3 = MAX(DP1+DP2,DP3,0) + 1.
On entry, if DP3 >= 0, then this array must contain the
coefficients of P3(x) in increasing powers of x.
On entry, if DP3 = -1, then P3(x) is taken to be the zero
polynomial.
On exit, the leading (DP3+1) elements of this array
contain the coefficients of P(x) in increasing powers of x
unless DP3 = -1 on exit, in which case the coefficients of
P(x) (the zero polynomial) are not stored in the array.
This is the case, for instance, when ALPHA = 0.0 and
P1(x) or P2(x) is the zero polynomial.

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

```
Method
```  Given real polynomials

DP1           i           DP2           i
P1(x) = SUM a(i+1) * x ,  P2(x) = SUM b(i+1) * x  and
i=0                       i=0

DP3           i
P3(x) = SUM c(i+1) * x ,
i=0

the routine computes the coefficents of P(x) = P1(x) * P2(x) +
DP3            i
alpha * P3(x) = SUM  d(i+1) * x  as follows.
i=0

Let e(i) = c(i) for 1 <= i <= DP3+1 and e(i) = 0 for i > DP3+1.
Then if DP1 >= DP2,

i
d(i) = SUM a(k) * b(i-k+1) + f(i), for i = 1, ..., DP2+1,
k=1

i
d(i)  = SUM a(k) * b(i-k+1) + f(i), for i = DP2+2, ..., DP1+1
k=i-DP2

and
DP1+1
d(i)  = SUM a(k) * b(i-k+1) + f(i) for i = DP1+2,...,DP1+DP2+1,
k=i-DP2

where f(i) = alpha * e(i).

Similar formulas hold for the case DP1 < DP2.

```
Numerical Aspects
```  None.

```
```  None
```
Example

Program Text

```*     MC01RD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          DP1MAX, DP2MAX, DP3MAX
PARAMETER        ( DP1MAX = 10, DP2MAX = 10, DP3MAX = 10 )
INTEGER          LENP3
PARAMETER        ( LENP3 = MAX(DP1MAX+DP2MAX,DP3MAX)+1 )
*     .. Local Scalars ..
DOUBLE PRECISION ALPHA
INTEGER          DP1, DP2, DP3, I, INFO
*     .. Local Arrays ..
DOUBLE PRECISION P1(DP1MAX+1), P2(DP2MAX+1), P3(LENP3)
*    \$                 P3(DP1MAX+DP2MAX+DP3MAX+1)
*     .. External Subroutines ..
EXTERNAL         MC01RD
*     .. Intrinsic Functions ..
INTRINSIC        MAX
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) DP1
IF ( DP1.LE.-2 .OR. DP1.GT.DP1MAX ) THEN
WRITE ( NOUT, FMT = 99994 ) DP1
ELSE
READ ( NIN, FMT = * ) ( P1(I), I = 1,DP1+1 )
READ ( NIN, FMT = * ) DP2
IF ( DP2.LE.-2 .OR. DP2.GT.DP2MAX ) THEN
WRITE ( NOUT, FMT = 99993 ) DP2
ELSE
READ ( NIN, FMT = * ) ( P2(I), I = 1,DP2+1 )
READ ( NIN, FMT = * ) DP3
IF ( DP3.LE.-2 .OR. DP3.GT.DP3MAX ) THEN
WRITE ( NOUT, FMT = 99992 ) DP3
ELSE
READ ( NIN, FMT = * ) ( P3(I), I = 1,DP3+1 )
END IF
READ ( NIN, FMT = * ) ALPHA
*           Compute the coefficients of the polynomial P(x).
CALL MC01RD( DP1, DP2, DP3, ALPHA, P1, P2, P3, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) DP3
IF ( DP3.GE.0 ) THEN
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 0, DP3
WRITE ( NOUT, FMT = 99995 ) I, P3(I+1)
20             CONTINUE
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MC01RD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01RD = ',I2)
99997 FORMAT (' Degree of the resulting polynomial P(x) = ',I2)
99996 FORMAT (/' The coefficients of P(x) are ',//' power of x     coe',
\$       'fficient ')
99995 FORMAT (2X,I5,9X,F9.4)
99994 FORMAT (/' DP1 is out of range.',/' DP1 = ',I5)
99993 FORMAT (/' DP2 is out of range.',/' DP2 = ',I5)
99992 FORMAT (/' DP3 is out of range.',/' DP3 = ',I5)
END
```
Program Data
``` MC01RD EXAMPLE PROGRAM DATA
1
1.00  2.50
2
1.00  0.10  -0.40
1
1.15  1.50
-2.20
```
Program Results
``` MC01RD EXAMPLE PROGRAM RESULTS

Degree of the resulting polynomial P(x) =  3

The coefficients of P(x) are

power of x     coefficient
0           -1.5300
1           -0.7000
2           -0.1500
3           -1.0000
```