**Purpose**

To compute, for two given real polynomials A(x) and B(x), the quotient polynomial Q(x) and the remainder polynomial R(x) of A(x) divided by B(x). The polynomials Q(x) and R(x) satisfy the relationship A(x) = B(x) * Q(x) + R(x), where the degree of R(x) is less than the degree of B(x).

SUBROUTINE MC01QD( DA, DB, A, B, RQ, IWARN, INFO ) C .. Scalar Arguments .. INTEGER DA, DB, INFO, IWARN C .. Array Arguments .. DOUBLE PRECISION A(*), B(*), RQ(*)

**Input/Output Parameters**

DA (input) INTEGER The degree of the numerator polynomial A(x). DA >= -1. DB (input/output) INTEGER On entry, the degree of the denominator polynomial B(x). DB >= 0. On exit, if B(DB+1) = 0.0 on entry, then DB contains the index of the highest power of x for which B(DB+1) <> 0.0. A (input) DOUBLE PRECISION array, dimension (DA+1) This array must contain the coefficients of the numerator polynomial A(x) in increasing powers of x unless DA = -1 on entry, in which case A(x) is taken to be the zero polynomial. B (input) DOUBLE PRECISION array, dimension (DB+1) This array must contain the coefficients of the denominator polynomial B(x) in increasing powers of x. RQ (output) DOUBLE PRECISION array, dimension (DA+1) If DA < DB on exit, then this array contains the coefficients of the remainder polynomial R(x) in increasing powers of x; Q(x) is the zero polynomial. Otherwise, the leading DB elements of this array contain the coefficients of R(x) in increasing powers of x, and the next (DA-DB+1) elements contain the coefficients of Q(x) in increasing powers of x.

IWARN INTEGER = 0: no warning; = k: if the degree of the denominator polynomial B(x) has been reduced to (DB - k) because B(DB+1-j) = 0.0 on entry for j = 0, 1, ..., k-1 and B(DB+1-k) <> 0.0.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: if on entry, DB >= 0 and B(i) = 0.0, where i = 1, 2, ..., DB+1.

Given real polynomials DA A(x) = a(1) + a(2) * x + ... + a(DA+1) * x and DB B(x) = b(1) + b(2) * x + ... + b(DB+1) * x where b(DB+1) is non-zero, the routine computes the coeffcients of the quotient polynomial DA-DB Q(x) = q(1) + q(2) * x + ... + q(DA-DB+1) * x and the remainder polynomial DB-1 R(x) = r(1) + r(2) * x + ... + r(DB) * x such that A(x) = B(x) * Q(x) + R(x). The algorithm used is synthetic division of polynomials (see [1]), which involves the following steps: (a) compute q(k+1) = a(DB+k+1) / b(DB+1) and (b) set a(j) = a(j) - q(k+1) * b(j-k) for j = k+1, ..., DB+k. Steps (a) and (b) are performed for k = DA-DB, DA-DB-1, ..., 0 and the algorithm terminates with r(i) = a(i) for i = 1, 2, ..., DB.

[1] Knuth, D.E. The Art of Computer Programming, (Vol. 2, Seminumerical Algorithms). Addison-Wesley, Reading, Massachusetts (2nd Edition), 1981.

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**Program Text**

* MC01QD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER DAMAX, DBMAX PARAMETER ( DAMAX = 10, DBMAX = 10 ) * .. Local Scalars .. INTEGER DA, DB, DBB, DQ, DR, I, IMAX, INFO, IWARN * .. Local Arrays .. DOUBLE PRECISION A(DAMAX+1), B(DBMAX+1), RQ(DAMAX+1) * .. External Subroutines .. EXTERNAL MC01QD * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) DA IF ( DA.LE.-2 .OR. DA.GT.DAMAX ) THEN WRITE ( NOUT, FMT = 99991 ) DA ELSE READ ( NIN, FMT = * ) ( A(I), I = 1,DA+1 ) READ ( NIN, FMT = * ) DB DBB = DB IF ( DB.LE.-1 .OR. DB.GT.DBMAX ) THEN WRITE ( NOUT, FMT = 99990 ) DB ELSE READ ( NIN, FMT = * ) ( B(I), I = 1,DB+1 ) * Compute Q(x) and R(x) from the given A(x) and B(x). CALL MC01QD( DA, DB, A, B, RQ, IWARN, INFO ) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE IF ( IWARN.NE.0 ) THEN WRITE ( NOUT, FMT = 99997 ) IWARN WRITE ( NOUT, FMT = 99996 ) DBB, DB END IF WRITE ( NOUT, FMT = 99995 ) DQ = DA - DB DR = DB - 1 IMAX = DQ IF ( DR.GT.IMAX ) IMAX = DR DO 20 I = 0, IMAX IF ( I.LE.DQ .AND. I.LE.DR ) THEN WRITE ( NOUT, FMT = 99994 ) I, RQ(DB+I+1), RQ(I+1) ELSE IF ( I.LE.DQ ) THEN WRITE ( NOUT, FMT = 99993 ) I, RQ(DB+I+1) ELSE WRITE ( NOUT, FMT = 99992 ) I, RQ(I+1) END IF 20 CONTINUE END IF END IF END IF * STOP * 99999 FORMAT (' MC01QD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from MC01QD = ',I2) 99997 FORMAT (' IWARN on exit from MC01QD = ',I2,/) 99996 FORMAT (' The degree of the denominator polynomial B(x) has been', $ ' reduced from ',I2,' to ',I2,/) 99995 FORMAT (' The coefficients of the polynomials Q(x) and R(x) are ', $ //' Q(x) R(x) ',/' power of', $ ' x coefficient coefficient ') 99994 FORMAT (2X,I5,9X,F9.4,7X,F9.4) 99993 FORMAT (2X,I5,9X,F9.4) 99992 FORMAT (2X,I5,25X,F9.4) 99991 FORMAT (/' DA is out of range.',/' DA = ',I5) 99990 FORMAT (/' DB is out of range.',/' DB = ',I5) END

MC01QD EXAMPLE PROGRAM DATA 4 2.0 2.0 -1.0 2.0 1.0 2 1.0 -1.0 1.0

MC01QD EXAMPLE PROGRAM RESULTS The coefficients of the polynomials Q(x) and R(x) are Q(x) R(x) power of x coefficient coefficient 0 1.0000 1.0000 1 3.0000 0.0000 2 1.0000