## 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.

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

```
Example

Program Text

```*     MC03MD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. 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
```