pm_polynomial::setPolyDiv Interface Reference

Return the quotient and remainder of dividing a polynomial with another polynomial of arbitrary degrees. More...

Detailed Description

Return the quotient and remainder of dividing a polynomial with another polynomial of arbitrary degrees.

See the documentation of pm_polynomial for details of the implementation.

Parameters

[in]	dividend	: The input contiguous vector of non-zero size of, type complex of kind any supported by the processor (e.g., CK, CK32, CK64, or CK128), or type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the coefficients of the dividend polynomial in the order of increasing power. By definition, the degree of the dividend polynomial is size(dividend) - 1. This means that the condition dividend(size(dividend)) /= 0. must hold.
[in]	divisor	: The input contiguous vector of non-zero size of the same type and kind as dividend, containing the coefficients of the divisor polynomial in the order of increasing power. By definition, the degree of the divisor polynomial is size(divisor) - 1. This means that the condition divisor(size(divisor)) /= 0. must hold.
[out]	quorem	: The output contiguous vector of the same type, kind and size as the input dividend, containing the coefficients of the quotient and remainder polynomials resulting of the polynomial division. The slice quorem(1 : lenQuo) contains the coefficients of the resulting quotient polynomial, in the order of increasing power. The slice quorem(lenQuo + 1 :) contains the coefficients of the resulting remainder polynomial, in the order of increasing power.
[out]	lenQuo	: The output scalar integer of default kind IK containing the length of the vector of coefficients of the resulting quotient. By definition, If the condition dividend == divisor holds, then lenQuo = size(dividend). If the condition size(dividend) < size(divisor) holds, then lenQuo = 0_IK. If the condition lenQuo == size(dividend) holds, then the remainder of the division is zero, as implied by the empty slice quorem(lenQuo + 1 : size(dividend)).

Possible calling interfaces ⛓

use pm_polynomial, only: setPolyDiv
 
call setPolyDiv(dividend(:), divisor(:), quorem(1:size(dividend)), lenQuo)

Warning

The condition 0 < size(divisor) must hold for the corresponding input arguments.
The condition 0 < size(dividend) must hold for the corresponding input arguments.
The condition size(quorem) == size(dividend) must hold for the corresponding input arguments.
The condition dividend(size(dividend)) /= 0. must hold for the corresponding input arguments.
The condition divisor(size(divisor)) /= 0. must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1.
By default, these procedures are pure in release build and impure in debug and testing builds.

See also

getPolySub
setPolySub
getPolySub
setPolySub
setPolyDiv
getPolyMul
setPolyMul

Example usage ⛓

! Define example template macro to avoid duplications. See the example output for actual usage.
#define SET_POLY_DIV \
block; \
    TYPE(RKG), allocatable :: dividend(:), divisor(:), quorem(:); \
    dividend = DIVIDEND; \
    divisor = DIVISOR; \
    call disp%skip(); \
    call disp%show("getPolyStr(dividend)"); \
    call disp%show( getPolyStr(dividend) ); \
    call disp%show("getPolyStr(divisor)"); \
    call disp%show( getPolyStr(divisor) ); \
    call disp%show("allocate(quorem, mold = dividend)"); \
                    allocate(quorem, mold = dividend); \
    call disp%show("call setPolyDiv(dividend, divisor, quorem, lenQuo)"); \
                    call setPolyDiv(dividend, divisor, quorem, lenQuo); \
    call disp%show("lenQuo - 1 ! Degree of quotient."); \
    call disp%show( lenQuo - 1 ); \
    call disp%show("getPolyStr(quorem(1:lenQuo)) ! Quotient."); \
    call disp%show( getPolyStr(quorem(1:lenQuo)) ); \
    call disp%show("getPolyStr(quorem(lenQuo + 1 :)) ! Remainder."); \
    call disp%show( getPolyStr(quorem(lenQuo + 1 :)) ); \
    call disp%show("getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder."); \
    call disp%show( getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ); \
    call disp%show("getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder."); \
    call disp%show( getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ); \
    call disp%skip(); \
end block;
 
 
program example
 
    use pm_kind, only: SK, IK
    use pm_kind, only: RKG => RKS ! all processor real and complex kinds are supported.
    use pm_io, only: display_type
    use pm_polynomial, only: getPolyMul
    use pm_polynomial, only: getPolyAdd
    use pm_polynomial, only: setPolyDiv
    use pm_polynomial, only: getPolyStr
 
    implicit none
 
    integer(IK) :: lenQuo
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
#define DIVIDEND [real(RKG) :: -4., 0., -2., 1.]
#define DIVISOR [real(RKG) :: -3., 1.]
#define TYPE real
SET_POLY_DIV ! [3., 1., 1.], [5.]
 
#define DIVIDEND [real(RKG) :: 2., 3., 1.]
#define DIVISOR [real(RKG) :: 1., 1.]
#define TYPE real
SET_POLY_DIV
 
#define DIVIDEND [real(RKG) :: -42., 0., -12., 1.]
#define DIVISOR [real(RKG) :: 1., -2., 1.]
#define TYPE real
SET_POLY_DIV ! [-32, -21]
 
#define DIVIDEND [real(RKG) :: -42., 0., -12., 1.]
#define DIVISOR [real(RKG) :: -2., 1.]
#define TYPE real
SET_POLY_DIV
 
#define DIVIDEND cmplx([real(RKG) :: -4., 0., -2., 1.], -[real(RKG) :: -4., 0., -2., 1.], RKG)
#define DIVISOR cmplx([real(RKG) :: -3., 1.], -[real(RKG) :: -3., 1.], RKG)
#define TYPE complex
SET_POLY_DIV
 
#define DIVIDEND cmplx([real(RKG) :: 2., 3., 1.], -[real(RKG) :: 2., 3., 1.], RKG)
#define DIVISOR cmplx([real(RKG) :: 1., 1.], -[real(RKG) :: 1., 1.], RKG)
#define TYPE complex
SET_POLY_DIV
 
#define DIVIDEND cmplx([real(RKG) :: -42., 0., -12., 1.], -[real(RKG) :: -42., 0., -12., 1.], RKG)
#define DIVISOR cmplx([real(RKG) :: 1., -2., 1.], -[real(RKG) :: 1., -2., 1.], RKG)
#define TYPE complex
SET_POLY_DIV
 
#define DIVIDEND cmplx([real(RKG) :: -42., 0., -12., 1.], -[real(RKG) :: -42., 0., -12., 1.], RKG)
#define DIVISOR cmplx([real(RKG) :: -2., 1.], -[real(RKG) :: -2., 1.])
#define TYPE complex
SET_POLY_DIV
 
end program example

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example output ⛓

 
getPolyStr(dividend)
-4.x^0 + 0.x^1 - 2.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                              
getPolyStr(divisor)
-3.x^0 + 1.x^1                                                                                                                                                                                                                                
allocate(quorem, mold = dividend)
call setPolyDiv(dividend, divisor, quorem, lenQuo)
lenQuo - 1 ! Degree of quotient.
+2
getPolyStr(quorem(1:lenQuo)) ! Quotient.
3.x^0 + 1.x^1 + 1.x^2                                                                                                                                                                                                                                                                                                                                                
getPolyStr(quorem(lenQuo + 1 :)) ! Remainder.
5.x^0                                                                                                                  
getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
-9.x^0 + 0.x^1 - 2.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                              
getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
-4.x^0 + 0.x^1 - 2.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                              
 
 
getPolyStr(dividend)
2.x^0 + 3.x^1 + 1.x^2                                                                                                                                                                                                                                                                                                                                                
getPolyStr(divisor)
1.x^0 + 1.x^1                                                                                                                                                                                                                                 
allocate(quorem, mold = dividend)
call setPolyDiv(dividend, divisor, quorem, lenQuo)
lenQuo - 1 ! Degree of quotient.
+1
getPolyStr(quorem(1:lenQuo)) ! Quotient.
2.x^0 + 1.x^1                                                                                                                                                                                                                                 
getPolyStr(quorem(lenQuo + 1 :)) ! Remainder.
0.x^0                                                                                                                  
getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
2.x^0 + 3.x^1 + 1.x^2                                                                                                                                                                                                                                                                                                                                                
getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
2.x^0 + 3.x^1 + 1.x^2                                                                                                                                                                                                                                                                                                                                                
 
 
getPolyStr(dividend)
-42.x^0 + 0.x^1 - 12.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                              
getPolyStr(divisor)
1.x^0 - 2.x^1 + 1.x^2                                                                                                                                                                                                                                                                                                                                                
allocate(quorem, mold = dividend)
call setPolyDiv(dividend, divisor, quorem, lenQuo)
lenQuo - 1 ! Degree of quotient.
+1
getPolyStr(quorem(1:lenQuo)) ! Quotient.
-10.x^0 + 1.x^1                                                                                                                                                                                                                                
getPolyStr(quorem(lenQuo + 1 :)) ! Remainder.
-32.x^0 - 21.x^1                                                                                                                                                                                                                                
getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
-10.x^0 + 21.x^1 - 12.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                              
getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
-42.x^0 + 0.x^1 - 12.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                              
 
 
getPolyStr(dividend)
-42.x^0 + 0.x^1 - 12.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                              
getPolyStr(divisor)
-2.x^0 + 1.x^1                                                                                                                                                                                                                                
allocate(quorem, mold = dividend)
call setPolyDiv(dividend, divisor, quorem, lenQuo)
lenQuo - 1 ! Degree of quotient.
+2
getPolyStr(quorem(1:lenQuo)) ! Quotient.
-20.x^0 - 10.x^1 + 1.x^2                                                                                                                                                                                                                                                                                                                                               
getPolyStr(quorem(lenQuo + 1 :)) ! Remainder.
-82.x^0                                                                                                                 
getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
40.x^0 + 0.x^1 - 12.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                               
getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
-42.x^0 + 0.x^1 - 12.x^2 + 1.x^3                                                                                                                                                                                                                                                                                                                                                                                                                                                              
 
 
getPolyStr(dividend)
(-4.,4.)x^0 + (0.,-0.)x^1 + (-2.,2.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                       
getPolyStr(divisor)
(-3.,3.)x^0 + (1.,-1.)x^1                                                                                                                                                                                                     
allocate(quorem, mold = dividend)
call setPolyDiv(dividend, divisor, quorem, lenQuo)
lenQuo - 1 ! Degree of quotient.
+2
getPolyStr(quorem(1:lenQuo)) ! Quotient.
(3.,0.)x^0 + (1.,0.)x^1 + (1.,0.)x^2                                                                                                                                                                                                                                                                                                         
getPolyStr(quorem(lenQuo + 1 :)) ! Remainder.
(5.,0.)x^0                                                                                                     
getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
(-9.,9.)x^0 + (0.,0.)x^1 + (-2.,2.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                        
getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
(-4.,9.)x^0 + (0.,0.)x^1 + (-2.,2.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                        
 
 
getPolyStr(dividend)
(2.,-2.)x^0 + (3.,-3.)x^1 + (1.,-1.)x^2                                                                                                                                                                                                                                                                                                      
getPolyStr(divisor)
(1.,-1.)x^0 + (1.,-1.)x^1                                                                                                                                                                                                     
allocate(quorem, mold = dividend)
call setPolyDiv(dividend, divisor, quorem, lenQuo)
lenQuo - 1 ! Degree of quotient.
+1
getPolyStr(quorem(1:lenQuo)) ! Quotient.
(2.,0.)x^0 + (1.,0.)x^1                                                                                                                                                                                                       
getPolyStr(quorem(lenQuo + 1 :)) ! Remainder.
(0.,0.)x^0                                                                                                     
getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
(2.,-2.)x^0 + (3.,-3.)x^1 + (1.,-1.)x^2                                                                                                                                                                                                                                                                                                      
getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
(2.,-2.)x^0 + (3.,-3.)x^1 + (1.,-1.)x^2                                                                                                                                                                                                                                                                                                      
 
 
getPolyStr(dividend)
(-42.,42.)x^0 + (0.,-0.)x^1 + (-12.,12.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                       
getPolyStr(divisor)
(1.,-1.)x^0 + (-2.,2.)x^1 + (1.,-1.)x^2                                                                                                                                                                                                                                                                                                      
allocate(quorem, mold = dividend)
call setPolyDiv(dividend, divisor, quorem, lenQuo)
lenQuo - 1 ! Degree of quotient.
+1
getPolyStr(quorem(1:lenQuo)) ! Quotient.
(-10.,0.)x^0 + (1.,0.)x^1                                                                                                                                                                                                      
getPolyStr(quorem(lenQuo + 1 :)) ! Remainder.
(-32.,32.)x^0 + (-21.,21.)x^1                                                                                                                                                                                                     
getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
(-10.,10.)x^0 + (21.,-21.)x^1 + (-12.,12.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                       
getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
(-42.,42.)x^0 + (0.,0.)x^1 + (-12.,12.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                        
 
 
getPolyStr(dividend)
(-42.,42.)x^0 + (0.,-0.)x^1 + (-12.,12.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                       
getPolyStr(divisor)
(-2.,2.)x^0 + (1.,-1.)x^1                                                                                                                                                                                                     
allocate(quorem, mold = dividend)
call setPolyDiv(dividend, divisor, quorem, lenQuo)
lenQuo - 1 ! Degree of quotient.
+2
getPolyStr(quorem(1:lenQuo)) ! Quotient.
(-20.,0.)x^0 + (-10.,0.)x^1 + (1.,0.)x^2                                                                                                                                                                                                                                                                                                       
getPolyStr(quorem(lenQuo + 1 :)) ! Remainder.
(-82.,0.)x^0                                                                                                    
getPolyStr(getPolyMul(divisor, quorem(1:lenQuo))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
(40.,-40.)x^0 + (0.,0.)x^1 + (-12.,12.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                        
getPolyStr(getPolyAdd(quorem(lenQuo + 1 :), getPolyMul(divisor, quorem(1:lenQuo)))) ! Reconstruct the dividend from the divisor, Quotient, and the Remainder.
(-42.,-40.)x^0 + (0.,0.)x^1 + (-12.,12.)x^2 + (1.,-1.)x^3                                                                                                                                                                                                                                                                                                                                                                                                       
 
 

Test:

test_pm_polynomial

Final Remarks ⛓

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If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
   
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This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Copyright

Computational Data Science Lab

Author:

Fatemeh Bagheri, Tuesday 11:34 PM, August 10, 2021, Dallas, TX

Definition at line 2209 of file pm_polynomial.F90.

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The documentation for this interface was generated from the following file:

• src/fortran/main/pm_polynomial.F90