Generate and return the roots of a polynomial of arbitrary degree specified by its coefficients coef.
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Detailed Description

Generate and return the roots of a polynomial of arbitrary degree specified by its coefficients coef.

See the documentation of pm_polynomial for details of the root-finding methods.

Parameters

[in]

coef

: The input contiguous vector 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 polynomial in the order of increasing power.
By definition, the degree of the polynomial is size(coef) - 1.

[in]

method

: The input scalar constant that can be any of the following:

the scalar constant eigen or a constant object of type eigen_type implying the use of the Eigenvalue root-finding method.
the scalar constant jenkins or a constant object of type jenkins_type implying the use of the Jenkins-Traub root-finding method.
the scalar constant laguerre or a constant object of type laguerre_type implying the use of the Laguerre root-finding method.
the scalar constant sgl or a constant object of type sgl_type implying the use of the Skowron-Gould root-finding method.
This method is yet to be fully implemented.

Which polynomial root-finding method should I use?

If you have a polynomial of highly varying coefficients, then the Eigenvalue method as specified by eigen_type is likely more reliable.
The Jenkins-Traub is also considered a relatively reliable fast Sure-Fire technique for finding the roots of polynomials.

(optional, default = eigen_type)

Returns: root : The output allocatable vector of type complex of the same kind as the input coef, containing the roots of the polynomial specified by its coefficients coef.
By definition, the size of the output root is the same as the degree of the polynomial (i.e., size(coef) - 1).
However, if the algorithm fails to find any of the roots at any stage, it will return only the roots found.
An output root of zero size indicates the failure of the algorithm to converge or find any of the polynomial roots.

Possible calling interfaces ⛓

: use pm_polynomial, only: getPolyRoot, eigen, jenkins, laguerre, sgl

complex(kind(coef)), allocatable :: root(:)

root = getPolyRoot(coef(:)) ! allocatable output.

root = getPolyRoot(coef(:), method) ! allocatable output.

pm_polynomial::getPolyRoot
Generate and return the roots of a polynomial of arbitrary degree specified by its coefficients coef.
Definition: pm_polynomial.F90:3750

pm_polynomial
This module contains procedures and generic interfaces for performing various mathematical operations...
Definition: pm_polynomial.F90:437

pm_polynomial::eigen
type(eigen_type), parameter eigen
This is a scalar parameter object of type eigen_type that is exclusively used to signify the use of E...
Definition: pm_polynomial.F90:3337

pm_polynomial::jenkins
type(jenkins_type), parameter jenkins
This is a scalar parameter object of type jenkins_type that is exclusively used to signify the use of...
Definition: pm_polynomial.F90:3399

pm_polynomial::laguerre
type(laguerre_type), parameter laguerre
This is a scalar parameter object of type laguerre_type that is exclusively used to signify the use o...
Definition: pm_polynomial.F90:3461

pm_polynomial::sgl
type(sgl_type), parameter sgl
This is a scalar parameter object of type sgl_type that is exclusively used to signify the use of Sko...
Definition: pm_polynomial.F90:3275

Warning: All warnings and conditions associated with setPolyRoot also apply to this generic interface.
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. The procedures of this generic interface are always impure when the input argument method is set to an object of type eigen_type.

See also: getRoot
setRoot
getPolyRoot
setPolyRoot

Example usage ⛓

: 1program example

2

3 use pm_kind, only: SK, IK, LK

4 use pm_kind, only: CKG => CKS ! all processor real and complex kinds are supported.

5 use pm_io, only: display_type

6 use pm_polynomial, only: getPolyRoot, getPolyVal

7 use pm_polynomial, only: eigen, jenkins, laguerre

8

9 implicit none

10

11 complex(CKG), allocatable :: coef(:), root(:)

12

13 type(display_type) :: disp

14 disp = display_type(file = "main.out.F90")

15

16 call disp%skip()

17 call disp%show("coef = [complex(CKG) :: -4, 1, -4, 1]")

18 coef = [complex(CKG) :: -4, 1, -4, 1]

19 call disp%show("coef")

20 call disp%show( coef )

21 call disp%show("root = getPolyRoot(coef%re) ! polynomial with real coefficients.")

22 root = getPolyRoot(coef%re)

23 call disp%show("root")

24 call disp%show( root )

25 call disp%show("getPolyVal(coef, root)")

26 call disp%show( getPolyVal(coef, root) )

27 call disp%skip()

28 call disp%show("root = getPolyRoot(coef%re, method = eigen) ! polynomial with real coefficients.")

29 root = getPolyRoot(coef%re, method = eigen)

30 call disp%show("root")

31 call disp%show( root )

32 call disp%show("getPolyVal(coef, root)")

33 call disp%show( getPolyVal(coef, root) )

34 call disp%skip()

35 call disp%show("root = getPolyRoot(coef%re, method = jenkins) ! polynomial with real coefficients.")

36 root = getPolyRoot(coef%re, method = jenkins)

37 call disp%show("root")

38 call disp%show( root )

39 call disp%show("getPolyVal(coef, root)")

40 call disp%show( getPolyVal(coef, root) )

41 call disp%skip()

42 call disp%show("root = getPolyRoot(coef%re, method = laguerre) ! polynomial with real coefficients.")

43 root = getPolyRoot(coef%re, method = laguerre)

44 call disp%show("root")

45 call disp%show( root )

46 call disp%show("getPolyVal(coef, root)")

47 call disp%show( getPolyVal(coef, root) )

48 call disp%skip()

49

50 call disp%skip()

51 call disp%show("coef = [complex(CKG) :: -120, 274, -225, 85, -15, 1]")

52 coef = [complex(CKG) :: -120, 274, -225, 85, -15, 1]

53 call disp%show("coef")

54 call disp%show( coef )

55 call disp%show("root = getPolyRoot(coef%re) ! polynomial with real coefficients.")

56 root = getPolyRoot(coef%re)

57 call disp%show("root")

58 call disp%show( root )

59 call disp%show("getPolyVal(coef, root)")

60 call disp%show( getPolyVal(coef, root) )

61 call disp%skip()

62 call disp%show("root = getPolyRoot(coef%re, method = eigen) ! polynomial with real coefficients.")

63 root = getPolyRoot(coef%re, method = eigen)

64 call disp%show("root")

65 call disp%show( root )

66 call disp%show("getPolyVal(coef, root)")

67 call disp%show( getPolyVal(coef, root) )

68 call disp%skip()

69 call disp%show("root = getPolyRoot(coef%re, method = jenkins) ! polynomial with real coefficients.")

70 root = getPolyRoot(coef%re, method = jenkins)

71 call disp%show("root")

72 call disp%show( root )

73 call disp%show("getPolyVal(coef, root)")

74 call disp%show( getPolyVal(coef, root) )

75 call disp%skip()

76 call disp%show("root = getPolyRoot(coef%re, method = laguerre) ! polynomial with real coefficients.")

77 root = getPolyRoot(coef%re, method = laguerre)

78 call disp%show("root")

79 call disp%show( root )

80 call disp%show("getPolyVal(coef, root)")

81 call disp%show( getPolyVal(coef, root) )

82 call disp%skip()

83

84 call disp%skip()

85 call disp%show("coef = [complex(CKG) :: 8, -8, 16, -16, 8, -8]")

86 coef = [complex(CKG) :: 8, -8, 16, -16, 8, -8]

87 call disp%show("coef")

88 call disp%show( coef )

89 call disp%show("root = getPolyRoot(coef%re) ! polynomial with real coefficients.")

90 root = getPolyRoot(coef%re)

91 call disp%show("root")

92 call disp%show( root )

93 call disp%show("getPolyVal(coef, root)")

94 call disp%show( getPolyVal(coef, root) )

95 call disp%skip()

96 call disp%show("root = getPolyRoot(coef%re, method = eigen) ! polynomial with real coefficients.")

97 root = getPolyRoot(coef%re, method = eigen)

98 call disp%show("root")

99 call disp%show( root )

100 call disp%show("getPolyVal(coef, root)")

101 call disp%show( getPolyVal(coef, root) )

102 call disp%skip()

103 call disp%show("root = getPolyRoot(coef%re, method = jenkins) ! polynomial with real coefficients.")

104 root = getPolyRoot(coef%re, method = jenkins)

105 call disp%show("root")

106 call disp%show( root )

107 call disp%show("getPolyVal(coef, root)")

108 call disp%show( getPolyVal(coef, root) )

109 call disp%skip()

110 call disp%show("root = getPolyRoot(coef%re, method = laguerre) ! polynomial with real coefficients.")

111 root = getPolyRoot(coef%re, method = laguerre)

112 call disp%show("root")

113 call disp%show( root )

114 call disp%show("getPolyVal(coef, root)")

115 call disp%show( getPolyVal(coef, root) )

116 call disp%skip()

117

118 call disp%skip()

119 call disp%show("coef = [complex(CKG) :: -120, 274, -225, 85, -15, 1]")

120 coef = [complex(CKG) :: -120, 274, -225, 85, -15, 1]

121 call disp%show("coef")

122 call disp%show( coef )

123 call disp%show("root = getPolyRoot(coef)")

124 root = getPolyRoot(coef)

125 call disp%show("root")

126 call disp%show( root )

127 call disp%show("getPolyVal(coef, root)")

128 call disp%show( getPolyVal(coef, root) )

129 call disp%skip()

130 call disp%show("root = getPolyRoot(coef, method = eigen)")

131 root = getPolyRoot(coef, method = eigen)

132 call disp%show("root")

133 call disp%show( root )

134 call disp%show("getPolyVal(coef, root)")

135 call disp%show( getPolyVal(coef, root) )

136 call disp%skip()

137 call disp%show("root = getPolyRoot(coef, method = jenkins)")

138 root = getPolyRoot(coef, method = jenkins)

139 call disp%show("root")

140 call disp%show( root )

141 call disp%show("getPolyVal(coef, root)")

142 call disp%show( getPolyVal(coef, root) )

143 call disp%skip()

144 call disp%show("root = getPolyRoot(coef, method = laguerre)")

145 root = getPolyRoot(coef, method = laguerre)

146 call disp%show("root")

147 call disp%show( root )

148 call disp%show("getPolyVal(coef, root)")

149 call disp%show( getPolyVal(coef, root) )

150 call disp%skip()

151

152end program example

pm_io::show
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726

pm_io::skip
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508

pm_polynomial::getPolyVal
Generate and return the value of the polynomial of arbitrary degree whose coefficients are specified ...
Definition: pm_polynomial.F90:517

pm_io
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252

pm_io::disp
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393

pm_kind
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268

pm_kind::LK
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
Definition: pm_kind.F90:541

pm_kind::CKS
integer, parameter CKS
The single-precision complex kind in Fortran mode. On most platforms, this is a 32-bit real kind.
Definition: pm_kind.F90:570

pm_kind::IK
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540

pm_kind::SK
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539

pm_io::display_type
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

Example Unix compile command via Intel ifort compiler ⛓
1#!/usr/bin/env sh

2rm main.exe

3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe

4./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓
1del main.exe

2set PATH=..\..\..\lib;%PATH%

3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe

4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓
1#!/usr/bin/env sh

2rm main.exe

3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe

4./main.exe

Example output ⛓
1

2coef = [complex(CKG) :: -4, 1, -4, 1]

3coef

4(-4.00000000, +0.00000000), (+1.00000000, +0.00000000), (-4.00000000, +0.00000000), (+1.00000000, +0.00000000)

5root = getPolyRoot(coef%re) ! polynomial with real coefficients.

6root

7(+3.99999952, +0.00000000), (+0.00000000, +0.999999821), (+0.00000000, -0.999999821)

8getPolyVal(coef, root)

9(-0.810623169E-5, +0.00000000), (-0.143051147E-5, +0.357627812E-6), (-0.143051147E-5, -0.357627812E-6)

10

11root = getPolyRoot(coef%re, method = eigen) ! polynomial with real coefficients.

12root

13(+3.99999952, +0.00000000), (+0.00000000, +0.999999821), (+0.00000000, -0.999999821)

14getPolyVal(coef, root)

15(-0.810623169E-5, +0.00000000), (-0.143051147E-5, +0.357627812E-6), (-0.143051147E-5, -0.357627812E-6)

16

17root = getPolyRoot(coef%re, method = jenkins) ! polynomial with real coefficients.

18root

19(-0.894069672E-6, +1.00000000), (-0.894069672E-6, -1.00000000), (+4.00000191, +0.00000000)

20getPolyVal(coef, root)

21(+0.190734863E-5, +0.715255919E-5), (+0.190734863E-5, -0.715255919E-5), (+0.324249268E-4, +0.00000000)

22

23root = getPolyRoot(coef%re, method = laguerre) ! polynomial with real coefficients.

24root

25(+0.977888703E-8, +1.00000000), (-0.467116479E-8, -1.00000000), (+0.160798663E-8, -1.00000000)

26getPolyVal(coef, root)

27(+0.00000000, -0.391155481E-7), (+0.00000000, -0.186846592E-7), (+0.00000000, +0.643194653E-8)

28

29

30coef = [complex(CKG) :: -120, 274, -225, 85, -15, 1]

31coef

32(-120.000000, +0.00000000), (+274.000000, +0.00000000), (-225.000000, +0.00000000), (+85.0000000, +0.00000000), (-15.0000000, +0.00000000), (+1.00000000, +0.00000000)

33root = getPolyRoot(coef%re) ! polynomial with real coefficients.

34root

35(+4.99990797, +0.00000000), (+4.00018120, +0.00000000), (+2.99988103, +0.00000000), (+2.00003195, +0.00000000), (+0.999997020, +0.00000000)

36getPolyVal(coef, root)

37(-0.205993652E-2, +0.00000000), (-0.915527344E-3, +0.00000000), (-0.411987305E-3, +0.00000000), (-0.190734863E-3, +0.00000000), (-0.839233398E-4, +0.00000000)

38

39root = getPolyRoot(coef%re, method = eigen) ! polynomial with real coefficients.

40root

41(+4.99990797, +0.00000000), (+4.00018120, +0.00000000), (+2.99988103, +0.00000000), (+2.00003195, +0.00000000), (+0.999997020, +0.00000000)

42getPolyVal(coef, root)

43(-0.205993652E-2, +0.00000000), (-0.915527344E-3, +0.00000000), (-0.411987305E-3, +0.00000000), (-0.190734863E-3, +0.00000000), (-0.839233398E-4, +0.00000000)

44

45root = getPolyRoot(coef%re, method = jenkins) ! polynomial with real coefficients.

46root

47(+0.999999523, +0.00000000), (+2.00003052, +0.00000000), (+2.99990487, +0.00000000), (+4.00010586, +0.00000000), (+4.99995899, +0.00000000)

48getPolyVal(coef, root)

49(-0.152587891E-4, +0.00000000), (-0.183105469E-3, +0.00000000), (-0.373840332E-3, +0.00000000), (-0.671386719E-3, +0.00000000), (-0.106048584E-2, +0.00000000)

50

51root = getPolyRoot(coef%re, method = laguerre) ! polynomial with real coefficients.

52root

53(+0.999993324, +0.00000000), (+2.00000691, +0.931322575E-9), (+1.99999917, +0.291038305E-10), (+0.999999583, +0.349245965E-9), (+1.00000000, +0.593718141E-8)

54getPolyVal(coef, root)

55(-0.175476074E-3, +0.00000000), (-0.457763672E-4, -0.558787860E-8), (+0.762939453E-5, -0.174622983E-9), (-0.228881836E-4, +0.838191383E-8), (+0.00000000, +0.142492354E-6)

56

57

58coef = [complex(CKG) :: 8, -8, 16, -16, 8, -8]

59coef

60(+8.00000000, +0.00000000), (-8.00000000, +0.00000000), (+16.0000000, +0.00000000), (-16.0000000, +0.00000000), (+8.00000000, +0.00000000), (-8.00000000, +0.00000000)

61root = getPolyRoot(coef%re) ! polynomial with real coefficients.

62root

63(+0.999999642, +0.00000000), (-0.429153442E-4, +1.00021696), (-0.429153442E-4, -1.00021696), (+0.427067280E-4, +0.999782741), (+0.427067280E-4, -0.999782741)

64getPolyVal(coef, root)

65(+0.114440918E-4, +0.00000000), (+0.238418579E-5, -0.804837327E-6), (+0.238418579E-5, +0.804837327E-6), (+0.190734863E-5, -0.566709787E-6), (+0.190734863E-5, +0.566709787E-6)

66

67root = getPolyRoot(coef%re, method = eigen) ! polynomial with real coefficients.

68root

69(+0.999999642, +0.00000000), (-0.429153442E-4, +1.00021696), (-0.429153442E-4, -1.00021696), (+0.427067280E-4, +0.999782741), (+0.427067280E-4, -0.999782741)

70getPolyVal(coef, root)

71(+0.114440918E-4, +0.00000000), (+0.238418579E-5, -0.804837327E-6), (+0.238418579E-5, +0.804837327E-6), (+0.190734863E-5, -0.566709787E-6), (+0.190734863E-5, +0.566709787E-6)

72

73root = getPolyRoot(coef%re, method = jenkins) ! polynomial with real coefficients.

74root

75(-0.339653343E-5, +1.00000632), (-0.339653343E-5, -1.00000632), (+0.339158305E-5, +0.999993563), (+0.339158305E-5, -0.999993563), (+1.00000000, +0.00000000)

76getPolyVal(coef, root)

77(+0.00000000, -0.469048246E-6), (+0.00000000, +0.469048246E-6), (+0.00000000, +0.430129148E-6), (+0.00000000, -0.430129148E-6), (+0.00000000, +0.00000000)

78

79root = getPolyRoot(coef%re, method = laguerre) ! polynomial with real coefficients.

80root

81(+0.111623813E-3, +0.999268234), (-0.395235606E-3, -0.999610364), (+0.317461498E-3, +0.999678612), (+1.00000000, -0.395812094E-8), (+0.999999940, +0.162981451E-8)

82getPolyVal(coef, root)

83(+0.219345093E-4, -0.112091075E-4), (-0.953674316E-5, +0.959518366E-5), (+0.667572021E-5, +0.645453110E-5), (+0.00000000, +0.126659870E-6), (+0.143051147E-5, -0.521540500E-7)

84

85

86coef = [complex(CKG) :: -120, 274, -225, 85, -15, 1]

87coef

88(-120.000000, +0.00000000), (+274.000000, +0.00000000), (-225.000000, +0.00000000), (+85.0000000, +0.00000000), (-15.0000000, +0.00000000), (+1.00000000, +0.00000000)

89root = getPolyRoot(coef)

90root

91(+4.99997854, -0.120472396E-5), (+4.00004864, +0.389384786E-5), (+2.99997115, -0.376791718E-5), (+2.00000644, +0.113765248E-5), (+0.999999404, +0.159270712E-6)

92getPolyVal(coef, root)

93(-0.137329102E-3, -0.289111576E-4), (-0.129699707E-3, -0.233645478E-4), (-0.190734863E-3, -0.150715496E-4), (-0.762939453E-5, -0.682582322E-5), (-0.762939453E-5, +0.382250619E-5)

94

95root = getPolyRoot(coef, method = eigen)

96root

97(+4.99997854, -0.120472396E-5), (+4.00004864, +0.389384786E-5), (+2.99997115, -0.376791718E-5), (+2.00000644, +0.113765248E-5), (+0.999999404, +0.159270712E-6)

98getPolyVal(coef, root)

99(-0.137329102E-3, -0.289111576E-4), (-0.129699707E-3, -0.233645478E-4), (-0.190734863E-3, -0.150715496E-4), (-0.762939453E-5, -0.682582322E-5), (-0.762939453E-5, +0.382250619E-5)

100

101root = getPolyRoot(coef, method = jenkins)

102root

103(+1.00000048, -0.684522092E-7), (+2.00026107, -0.101812184E-3), (+2.99921584, +0.304903486E-3), (+4.00078106, -0.304773916E-3), (+4.99974155, +0.101751066E-3)

104getPolyVal(coef, root)

105(+0.762939453E-5, -0.164284984E-5), (-0.154876709E-2, +0.610602554E-3), (-0.317382812E-2, +0.121960416E-2), (-0.476837158E-2, +0.183103979E-2), (-0.605010986E-2, +0.243940251E-2)

106

107root = getPolyRoot(coef, method = laguerre)

108root

109(+0.999993324, +0.00000000), (+2.00000691, +0.931322575E-9), (+1.99999917, +0.291038305E-10), (+0.999999583, +0.349245965E-9), (+1.00000000, +0.593718141E-8)

110getPolyVal(coef, root)

111(-0.175476074E-3, +0.00000000), (-0.457763672E-4, -0.558787860E-8), (+0.762939453E-5, -0.174622983E-9), (-0.228881836E-4, +0.838191383E-8), (+0.00000000, +0.142492354E-6)

112

113

Benchmarks:

Benchmark :: The effects of method on runtime efficiency ⛓

: The following program compares the runtime performance of setPolyRoot using different polynomial root finding algorithms.

! Test the performance of `setPolyRoot()` with and without the selection `control` argument.
program benchmark
 
    use pm_bench, only: bench_type
    use pm_distUnif, only: setUnifRand
    use pm_arrayResize, only: setResized
    use pm_kind, only: SK, IK, LK, RKH, RK, RKG => RKD
    use iso_fortran_env, only: error_unit
 
    implicit none
 
    integer(IK)         , parameter     :: degmax = 9_IK
    integer(IK)         , allocatable   :: rootCount(:)
    integer(IK)                         :: ibench
    integer(IK)                         :: ideg
    integer(IK)                         :: degree
    integer(IK)                         :: fileUnit
    integer(IK)                         :: rootUnit
    real(RKG)                           :: dumsum = 0._RKG
    complex(RKG)                        :: coef(0:2**degmax)
    complex(RKG)                        :: root(2**degmax)
    type(bench_type)    , allocatable   :: bench(:)
 
    bench = [ bench_type(name = SK_"Eigen", exec = setPolyRootEigen, overhead = setOverhead) &
            , bench_type(name = SK_"Jenkins", exec = setPolyRootJenkins, overhead = setOverhead) &
            , bench_type(name = SK_"Laguerre", exec = setPolyRootLaguerre, overhead = setOverhead) &
            , bench_type(name = SK_"SGL", exec = setPolyRootSGL, overhead = setOverhead) &
            ]
    allocate(rootCount(size(bench)))
 
    write(*,"(*(g0,:,' '))")
    write(*,"(*(g0,:,' '))") "Benchmarking setPolyRoot()"
    write(*,"(*(g0,:,' '))")
 
    open(newunit = fileUnit, file = "main.out", status = "replace")
    open(newunit = rootUnit, file = "root.count", status = "replace")
 
        write(fileUnit, "(*(g0,:,','))") "Polynomial Degree", (bench(ibench)%name, ibench = 1, size(bench))
        write(rootUnit, "(*(g0,:,','))") "Polynomial Degree", (bench(ibench)%name, ibench = 1, size(bench))
        loopOverArraySize: do ideg = 0, degmax
 
            degree = 2**ideg
            call setUnifRand(coef(0 : degree), (1._RKG, 1._RKG), (2._RKG, 2._RKG))
            write(*,"(*(g0,:,' '))") "Benchmarking with coef size", degree
            do ibench = 1, size(bench)
                bench(ibench)%timing = bench(ibench)%getTiming(minsec = 0.07_RK)
            end do
            write(rootUnit,"(*(g0,:,','))") degree, rootCount
            write(fileUnit,"(*(g0,:,','))") degree, (bench(ibench)%timing%mean, ibench = 1, size(bench))
 
        end do loopOverArraySize
        write(*,"(*(g0,:,' '))") dumsum
        write(*,"(*(g0,:,' '))")
 
    close(fileUnit)
 
contains
 
    !%%%%%%%%%%%%%%%%%%%%
    ! procedure wrappers.
    !%%%%%%%%%%%%%%%%%%%%
 
    subroutine setOverhead()
        call getDummy()
    end subroutine
 
    subroutine getDummy()
        dumsum = dumsum + rootCount(ibench)
    end subroutine
 
    subroutine setPolyRootEigen()
        use pm_polynomial, only: setPolyRoot, method => eigen
        call setPolyRoot(root(1 : degree), rootCount(ibench), coef(0 : degree), method)
        call getDummy()
    end subroutine
 
    subroutine setPolyRootJenkins()
        use pm_polynomial, only: setPolyRoot, method => jenkins
        call setPolyRoot(root(1 : degree), rootCount(ibench), coef(0 : degree), method)
        call getDummy()
    end subroutine
 
    subroutine setPolyRootLaguerre()
        use pm_polynomial, only: setPolyRoot, method => laguerre
        call setPolyRoot(root(1 : degree), rootCount(ibench), coef(0 : degree), method)
        call getDummy()
    end subroutine
 
    subroutine setPolyRootSGL()
        use pm_polynomial, only: setPolyRoot, method => sgl
        call setPolyRoot(root(1 : degree), rootCount(ibench), coef(0 : degree), method)
        rootCount(ibench) = root(1)%re
        call getDummy()
    end subroutine
 
end program benchmark

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

Postprocessing of the benchmark output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
 
import os
dirname = os.path.basename(os.getcwd()) 
 
fontsize = 14
 
df = pd.read_csv("main.out", delimiter = ",")
colnames = list(df.columns.values)
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for colname in colnames[1:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
ax.set_title(" vs. ".join(colnames[1:])+"\nLower is better.", fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( colnames[1:]
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.png")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
plt.plot( df[colnames[0]].values
        , np.ones(len(df[colnames[0]].values))
        , linestyle = "--"
       #, color = "black"
        , linewidth = 2
        )
for colname in colnames[2:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values / df[colnames[1]].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
ax.set_title("Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}.".format(colnames[1]), fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( colnames[1:]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.ratio.png")
 
 
 
df = pd.read_csv("root.count", delimiter = ",")
colnames = list(df.columns.values)
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
plt.plot( range(1, df[colnames[0]].values[-1] * 2)
        , range(1, df[colnames[0]].values[-1] * 2)
        , linestyle = "--"
        , color = "black"
        , linewidth = 2
        )
for colname in colnames[1:-1]:
    plt.plot( df[colnames[0]].values
            , df[colname].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Number of Roots Found", fontsize = fontsize)
ax.set_title(" vs. ".join(colnames[1:])+"\nHigher is better.", fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( ["Equality"] + list(colnames[1:])
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".root.count.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

Among all summation algorithms, jenkins_type appears to offer the fastest root finding algorithms.
The eigen_type method also tends to offer excellent performance.
Unlike the above two, laguerre_type algorithm tends to significantly trail behind both in performance and reliability in finding all roots of the polynomial.

Test:: test_pm_polynomial

Final Remarks ⛓

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.

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.
If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

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:: Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 3750 of file pm_polynomial.F90.

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

src/fortran/main/pm_polynomial.F90