## MC03MD

### Real polynomial matrix operation P(x) = P1(x) P2(x) + alpha P3(x)

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

Purpose

      To compute the coefficients of the real polynomial matrix

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

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

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

Specification

          SUBROUTINE MC03MD( RP1, CP1, CP2, DP1, DP2, DP3, ALPHA, P1,
         $                   LDP11, LDP12, P2, LDP21, LDP22, P3, LDP31,
         $                   LDP32, DWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER           CP1, CP2, DP1, DP2, DP3, INFO, LDP11, LDP12,
         $                  LDP21, LDP22, LDP31, LDP32, RP1
          DOUBLE PRECISION  ALPHA
    C     .. Array Arguments ..
          DOUBLE PRECISION  DWORK(*), P1(LDP11,LDP12,*), P2(LDP21,LDP22,*),
         $                  P3(LDP31,LDP32,*)

Arguments

Input/Output Parameters

      RP1     (input) INTEGER
              The number of rows of the matrices P1(x) and P3(x).
              RP1 >= 0.

      CP1     (input) INTEGER
              The number of columns of matrix P1(x) and the number of
              rows of matrix P2(x).  CP1 >= 0.

      CP2     (input) INTEGER
              The number of columns of the matrices P2(x) and P3(x).
              CP2 >= 0.

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

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

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

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

      P1      (input) DOUBLE PRECISION array, dimension (LDP11,LDP12,*)
              If DP1 >= 0, then the leading RP1-by-CP1-by-(DP1+1) part
              of this array must contain the coefficients of the
              polynomial matrix P1(x). Specifically, P1(i,j,k) must
              contain the coefficient of x**(k-1) of the polynomial
              which is the (i,j)-th element of P1(x), where i = 1,2,...,
              RP1, j = 1,2,...,CP1 and k = 1,2,...,DP1+1.
              If DP1 = -1, then P1(x) is taken to be the zero polynomial
              matrix, P1 is not referenced and can be supplied as a
              dummy array (i.e. set the parameters LDP11 = LDP12 = 1 and
              declare this array to be P1(1,1,1) in the calling
              program).

      LDP11   INTEGER
              The leading dimension of array P1.
              LDP11 >= MAX(1,RP1) if DP1 >= 0,
              LDP11 >= 1          if DP1 = -1.

      LDP12   INTEGER
              The second dimension of array P1.
              LDP12 >= MAX(1,CP1) if DP1 >= 0,
              LDP12 >= 1          if DP1 = -1.

      P2      (input) DOUBLE PRECISION array, dimension (LDP21,LDP22,*)
              If DP2 >= 0, then the leading CP1-by-CP2-by-(DP2+1) part
              of this array must contain the coefficients of the
              polynomial matrix P2(x). Specifically, P2(i,j,k) must
              contain the coefficient of x**(k-1) of the polynomial
              which is the (i,j)-th element of P2(x), where i = 1,2,...,
              CP1, j = 1,2,...,CP2 and k = 1,2,...,DP2+1.
              If DP2 = -1, then P2(x) is taken to be the zero polynomial
              matrix, P2 is not referenced and can be supplied as a
              dummy array (i.e. set the parameters LDP21 = LDP22 = 1 and
              declare this array to be P2(1,1,1) in the calling
              program).

      LDP21   INTEGER
              The leading dimension of array P2.
              LDP21 >= MAX(1,CP1) if DP2 >= 0,
              LDP21 >= 1          if DP2 = -1.

      LDP22   INTEGER
              The second dimension of array P2.
              LDP22 >= MAX(1,CP2) if DP2 >= 0,
              LDP22 >= 1          if DP2 = -1.

      P3      (input/output) DOUBLE PRECISION array, dimension
              (LDP31,LDP32,n), where n = MAX(DP1+DP2,DP3,0)+1.
              On entry, if DP3 >= 0, then the leading
              RP1-by-CP2-by-(DP3+1) part of this array must contain the
              coefficients of the polynomial matrix P3(x). Specifically,
              P3(i,j,k) must contain the coefficient of x**(k-1) of the
              polynomial which is the (i,j)-th element of P3(x), where
              i = 1,2,...,RP1, j = 1,2,...,CP2 and k = 1,2,...,DP3+1.
              If DP3 = -1, then P3(x) is taken to be the zero polynomial
              matrix.
              On exit, if DP3 >= 0 on exit (ALPHA <> 0.0 and DP3 <> -1,
              on entry, or DP1 <> -1 and DP2 <> -1), then the leading
              RP1-by-CP2-by-(DP3+1) part of this array contains the
              coefficients of P(x). Specifically, P3(i,j,k) contains the
              coefficient of x**(k-1) of the polynomial which is the
              (i,j)-th element of P(x), where i = 1,2,...,RP1, j = 1,2,
              ...,CP2 and k = 1,2,...,DP3+1.
              If DP3 = -1 on exit, then the coefficients of P(x) (the
              zero polynomial matrix) are not stored in the array.

      LDP31   INTEGER
              The leading dimension of array P3.  LDP31 >= MAX(1,RP1).

      LDP32   INTEGER
              The second dimension of array P3.   LDP32 >= MAX(1,CP2).

Workspace

      DWORK   DOUBLE PRECISION array, dimension (CP1)

Error Indicator

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

Method

      Given real polynomial matrices

                 DP1            i
         P1(x) = SUM (A(i+1) * x ),
                 i=0

                 DP2            i
         P2(x) = SUM (B(i+1) * x ),
                 i=0

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

      and a real scalar alpha, the routine computes the coefficients
      d ,d ,..., of the polynomial matrix
       1  2

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

      from the formula

                  s
         d    =  SUM (A(k+1) * B(i-k+1)) + alpha * C(i+1),
          i+1    k=r

      where i = 0,1,...,DP1+DP2 and r and s depend on the value of i
      (e.g. if i <= DP1 and i <= DP2, then r = 0 and s = i).

Numerical Aspects

      None.

Further Comments

      Other elementary operations involving polynomial matrices can
      easily be obtained by calling the appropriate BLAS routine(s).

Example

Program Text

    *     MC03MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          CP1MAX, CP2MAX, DP1MAX, DP2MAX, DP3MAX, RP1MAX
          PARAMETER        ( CP1MAX = 10, CP2MAX = 10, DP1MAX = 10,
         $                   DP2MAX = 10, DP3MAX = 20, RP1MAX = 10 )
          INTEGER          LDP11, LDP12, LDP21, LDP22, LDP31, LDP32
          PARAMETER        ( LDP11 = RP1MAX, LDP12 = CP1MAX,
         $                   LDP21 = CP1MAX, LDP22 = CP2MAX,
         $                   LDP31 = RP1MAX, LDP32 = CP2MAX )
    *     .. Local Scalars ..
          DOUBLE PRECISION ALPHA
          INTEGER          CP1, CP2, DP1, DP2, DP3, I, INFO, J, K, RP1
    *     .. Local Arrays ..
          DOUBLE PRECISION DWORK(CP1MAX),
         $                 P1(LDP11,LDP12,DP1MAX+1),
         $                 P2(LDP21,LDP22,DP2MAX+1),
         $                 P3(LDP31,LDP32,DP3MAX+1)
    *     .. External Subroutines ..
          EXTERNAL         MC03MD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) RP1, CP1, CP2
          IF ( RP1.LT.0 .OR. RP1.GT.RP1MAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) RP1
          ELSE IF ( CP1.LT.0 .OR. CP1.GT.CP1MAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) CP1
          ELSE IF ( CP2.LT.0 .OR. CP2.GT.CP2MAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) CP2
          ELSE
             READ ( NIN, FMT = * ) DP1
             IF ( DP1.LE.-2 .OR. DP1.GT.DP1MAX ) THEN
                WRITE ( NOUT, FMT = 99992 ) DP1
             ELSE
                DO 40 K = 1, DP1 + 1
                   DO 20 J = 1, CP1
                      READ ( NIN, FMT = * ) ( P1(I,J,K), I = 1,RP1 )
       20          CONTINUE
       40       CONTINUE
                READ ( NIN, FMT = * ) DP2
                IF ( DP2.LE.-2 .OR. DP2.GT.DP2MAX ) THEN
                   WRITE ( NOUT, FMT = 99991 ) DP2
                ELSE
                   DO 80 K = 1, DP2 + 1
                      DO 60 J = 1, CP2
                         READ ( NIN, FMT = * ) ( P2(I,J,K), I = 1,CP1 )
       60             CONTINUE
       80          CONTINUE
                   READ ( NIN, FMT = * ) DP3
                   IF ( DP3.LE.-2 .OR. DP3.GT.DP3MAX ) THEN
                      WRITE ( NOUT, FMT = 99990 ) DP3
                   ELSE
                      DO 120 K = 1, DP3 + 1
                         DO 100 J = 1, CP2
                            READ ( NIN, FMT = * ) ( P3(I,J,K), I = 1,RP1 )
      100                CONTINUE
      120             CONTINUE
                      READ ( NIN, FMT = * ) ALPHA
    *                 Compute the coefficients of the polynomial matrix P(x)
                      CALL MC03MD( RP1, CP1, CP2, DP1, DP2, DP3, ALPHA, P1,
         $                         LDP11, LDP12, P2, LDP21, LDP22, P3,
         $                         LDP31, LDP32, DWORK, INFO )
    *
                      IF ( INFO.NE.0 ) THEN
                         WRITE ( NOUT, FMT = 99998 ) INFO
                      ELSE
                         WRITE ( NOUT, FMT = 99997 ) DP3,
         $                     ( I-1, I = 1,DP3+1 )
                         DO 160 I = 1, RP1
                            DO 140 J = 1, CP2
                               WRITE ( NOUT, FMT = 99996 ) I, J,
         $                       ( P3(I,J,K), K = 1,DP3+1 )
      140                   CONTINUE
      160                CONTINUE
                      END IF
                   END IF
                END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' MC03MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MC03MD = ',I2)
    99997 FORMAT (' The polynomial matrix P(x) (of degree ',I2,') is ',
         $       //' power of x         ',20I8)
    99996 FORMAT (/' element (',I2,',',I2,') is ',20(1X,F7.2))
    99995 FORMAT (/' RP1 is out of range.',/' RP1 = ',I5)
    99994 FORMAT (/' CP1 is out of range.',/' CP1 = ',I5)
    99993 FORMAT (/' CP2 is out of range.',/' CP2 = ',I5)
    99992 FORMAT (/' DP1 is out of range.',/' DP1 = ',I5)
    99991 FORMAT (/' DP2 is out of range.',/' DP2 = ',I5)
    99990 FORMAT (/' DP3 is out of range.',/' DP3 = ',I5)
          END

Program Data

     MC03MD EXAMPLE PROGRAM DATA
       3     2     2
       2
       1.0   0.0   3.0
       2.0  -1.0   2.0
      -2.0   4.0   9.0
       3.0   7.0  -2.0
       6.0   2.0  -3.0
       1.0   2.0   4.0
       1
       6.0   1.0
       1.0   7.0
      -9.0  -6.0
       7.0   8.0
       1
       1.0   1.0   0.0
       0.0   1.0   1.0
      -1.0   1.0   1.0
      -1.0  -1.0   1.0
       1.0

Program Results

     MC03MD EXAMPLE PROGRAM RESULTS

     The polynomial matrix P(x) (of degree  3) is

     power of x                0       1       2       3

     element ( 1, 1) is     9.00  -31.00   37.00  -60.00

     element ( 1, 2) is    15.00   41.00   23.00   50.00

     element ( 2, 1) is     0.00   38.00  -64.00  -30.00

     element ( 2, 2) is    -6.00   44.00  100.00   30.00

     element ( 3, 1) is    20.00   14.00  -83.00    3.00

     element ( 3, 2) is    18.00   33.00   72.00   11.00

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