## 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.

Further Comments

      None

Example

Program Text

    *     MC01RD EXAMPLE PROGRAM TEXT
    *
    *     .. 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

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