Return the roots of a polynomial of arbitrary degree specified by its coefficients coef
.
More...
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 method.
- Parameters
-
[out] | root | : The output contiguous vector of type complex of the same kind as the input argument coef and the same size as the degree of the polynomial (i.e., size(coef) - 1 ), containing the roots of the polynomial determined from the polynomial coefficients coef .
|
[out] | count | : The output scalar of type integer of default kind IK, containing the total number of roots computed and stored in the output vector slice root(1 : count) .
A value of count = size(root) implies a successful computation of all polynomial roots.
A value of count = 0 implies a total failure of the algorithm in finding any roots.
The condition count < size(root) occurs if either,
-
the algorithm fails to converge, or
-
the algorithm fails to identify all roots of the polynomial, or
-
the condition
coef(size(coef)) == 0. occurs (i.e., when the coefficient of the highest power of the polynomial is zero), or
-
the condition
size(coef) < 2 occurs (i.e., when the input polynomial is a constant).
|
[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.
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.
|
Possible calling interfaces ⛓
call setPolyRoot(root(
1 : degree), count, coef(
0 : degree), method) ! `degree` is the degree of the polynomial.
Return the roots of a polynomial of arbitrary degree specified by its coefficients coef.
This module contains procedures and generic interfaces for performing various mathematical operations...
- Warning
- The condition
1 < size(coef)
must hold for the corresponding input arguments.
The condition coef(size(coef)) /= 0.
must hold for the corresponding input arguments (i.e., the coefficient of the highest-degree term must be non-zero).
The condition all(shape(workspace) == size(coef) - 1)
must hold for the corresponding input arguments.
The condition size(root) == size(coef) - 1
must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1
.
-
It is crucial to keep in mind that computers use a fixed number of binary digits to represent floating-point numbers.
As such polynomials of high degree can be problematic for root-finding algorithms.
- See also
- getRoot
setRoot
getPolyRoot
setPolyRoot
Example usage ⛓
15 type(display_type) :: disp
25#define GET_ROOT_EIG(CREP, TYPE, RKG) \
28TYPE(RKG) ,
allocatable :: coef(:); \
29complex(RKG),
allocatable :: root(:); \
33call disp%show( coef ,
format = "(sp,"//getStr(CREP)
//"(g0,:,', '))"); \
34call disp%show(
"call setResized(root, size(coef, 1, IK) - 1_IK)"); \
36call disp%show(
"call setPolyRoot(root, count, coef, eigen)"); \
41call disp%show( root(
1:count) ,
format = "(sp,2(g0,:,', '))"); \
42call disp%show(
"getPolyVal(coef, root(1:count))"); \
54#define GET_ROOT_JEN(CREP, TYPE, RKG) \
57TYPE(RKG) ,
allocatable :: coef(:); \
58complex(RKG),
allocatable :: root(:); \
62call disp%show( coef ,
format = "(sp,"//getStr(CREP)
//"(g0,:,', '))"); \
63call disp%show(
"call setResized(root, size(coef, 1, IK) - 1_IK)"); \
65call disp%show(
"call setPolyRoot(root, count, coef, jenkins)"); \
70call disp%show( root(
1:count) ,
format = "(sp,2(g0,:,', '))"); \
71call disp%show(
"getPolyVal(coef, root(1:count))"); \
83#define GET_ROOT_LAG(CREP, TYPE, RKG) \
86TYPE(RKG) ,
allocatable :: coef(:); \
87complex(RKG),
allocatable :: root(:); \
91call disp%show( coef ,
format = "(sp,"//getStr(CREP)
//"(g0,:,', '))"); \
92call disp%show(
"call setResized(root, size(coef, 1, IK) - 1_IK)"); \
94call disp%show(
"call setPolyRoot(root, count, coef, laguerre)"); \
99call disp%show( root(
1:count) ,
format = "(sp,2(g0,:,', '))"); \
100call disp%show(
"getPolyVal(coef, root(1:count))"); \
112#define GET_ROOT_SKO(CREP, TYPE, RKG) \
115TYPE(RKG) ,
allocatable :: coef(:); \
116complex(RKG),
allocatable :: root(:); \
120call disp%show( coef ,
format = "(sp,"//getStr(CREP)
//"(g0,:,', '))"); \
121call disp%show(
"call setResized(root, size(coef, 1, IK) - 1_IK)"); \
123call disp%show(
"call cmplx_roots_gen(root, coef, size(coef) - 1, .true., .true.)"); \
124 call cmplx_roots_gen(root, coef,
size(coef)
- 1,
.true.,
.true.); \
126call disp%show( root(
1:count) ,
format = "(sp,2(g0,:,', '))"); \
127call disp%show(
"getPolyVal(coef, root(1:count))"); \
133 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
134 call disp%show(
"! Compute the roots of polynomials with real coefficients.")
135 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
139 [
8,
-8,
16,
-16,
8,
-8]
140 GET_ROOT_EIG(
1, real,
RKS)
141 GET_ROOT_EIG(
1, real,
RKD)
142 GET_ROOT_EIG(
1, real,
RKH)
143 GET_ROOT_JEN(
1, real,
RKS)
144 GET_ROOT_JEN(
1, real,
RKD)
145 GET_ROOT_JEN(
1, real,
RKH)
146 GET_ROOT_LAG(
1, real,
RKS)
147 GET_ROOT_LAG(
1, real,
RKD)
148 GET_ROOT_LAG(
1, real,
RKH)
151 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
152 call disp%show(
"! Compute the roots of polynomials with real coefficients.")
153 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
157 [
3628800,
-10628640,
12753576,
-8409500,
3416930,
-902055,
157773,
-18150,
1320,
-55,
1]
158 GET_ROOT_EIG(
1, real,
RKS)
159 GET_ROOT_EIG(
1, real,
RKD)
160 GET_ROOT_EIG(
1, real,
RKH)
161 GET_ROOT_JEN(
1, real,
RKS)
162 GET_ROOT_JEN(
1, real,
RKD)
163 GET_ROOT_JEN(
1, real,
RKH)
164 GET_ROOT_LAG(
1, real,
RKS)
165 GET_ROOT_LAG(
1, real,
RKD)
166 GET_ROOT_LAG(
1, real,
RKH)
169 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
170 call disp%show(
"! Compute the roots of polynomials with complex coefficients with zeros 1,2,...,10.")
171 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
175 [
3628800,
-10628640,
12753576,
-8409500,
3416930,
-902055,
157773,
-18150,
1320,
-55,
1]
176 GET_ROOT_EIG(
2,
complex, RKS)
177 GET_ROOT_EIG(
2,
complex, RKD)
178 GET_ROOT_EIG(
2,
complex, RKH)
179 GET_ROOT_JEN(
2,
complex, RKS)
180 GET_ROOT_JEN(
2,
complex, RKD)
181 GET_ROOT_JEN(
2,
complex, RKH)
182 GET_ROOT_LAG(
2,
complex, RKS)
183 GET_ROOT_LAG(
2,
complex, RKD)
184 GET_ROOT_LAG(
2,
complex, RKH)
187 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
188 call disp%show(
"! Compute the roots of polynomials with complex coefficients with zeros on imaginary axis.")
189 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
193 [(
0._RKH,
1._RKH), (
-10001.0001_RKH,
0._RKH), (
0._RKH,
-10001.0001_RKH), (
1._RKH,
0._RKH)]
194 GET_ROOT_EIG(
2,
complex, RKD)
195 GET_ROOT_EIG(
2,
complex, RKH)
196 GET_ROOT_JEN(
2,
complex, RKD)
197 GET_ROOT_JEN(
2,
complex, RKH)
198 GET_ROOT_LAG(
2,
complex, RKD)
199 GET_ROOT_LAG(
2,
complex, RKH)
202 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
203 call disp%show(
"! Compute the roots of polynomials with complex coefficients with zeros at 1+i,1/2*(1+i)....1/(2**-9)*(1+i).")
204 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
208[ (
0._RKH,
9.094947017729282e-13_RKH) \
209, (
-4.652065399568528e-10_RKH,
-4.652065399568528e-10_RKH) \
210, (
1.584803612786345e-7_RKH,
0._RKH) \
211, (
-1.154642632172909e-5_RKH,
1.154642632172909e-5_RKH) \
212, (
0._RKH,
-7.820779428584501e-4_RKH) \
213, (
1.271507365163416e-2_RKH,
1.271507365163416e-2_RKH) \
214, (
-.
2002119533717632e0_RKH,
0._RKH) \
215, (.
7567065954208374e0_RKH,
-7.567065954208374e-1_RKH) \
216, (
0._RKH,
2.658859252929688e0_RKH) \
217, (
-1.998046875_RKH,
-1.998046875_RKH) \
220GET_ROOT_EIG(
2,
complex, RKD)
221GET_ROOT_EIG(
2,
complex, RKH)
222GET_ROOT_JEN(
2,
complex, RKD)
223GET_ROOT_JEN(
2,
complex, RKH)
224GET_ROOT_LAG(
2,
complex, RKD)
225GET_ROOT_LAG(
2,
complex, RKH)
226GET_ROOT_SKO(
2,
complex, RKD)
229 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
230 call disp%show(
"! Compute the roots of polynomials with complex coefficients with multiple zeros.")
231 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
235[ (
288._RKH,
0._RKH) \
236, (
-1344._RKH,
504._RKH) \
237, (
2204._RKH,
-2352._RKH) \
238, (
-920._RKH,
4334._RKH) \
239, (
-1587._RKH,
-3836._RKH) \
240, (
2374._RKH,
1394._RKH) \
241, (
-1293._RKH,
200._RKH) \
242, (
284._RKH,
-334._RKH) \
243, (
3._RKH,
100._RKH) \
244, (
-10._RKH,
-10._RKH) \
247GET_ROOT_EIG(
2,
complex, RKS)
248GET_ROOT_EIG(
2,
complex, RKD)
249GET_ROOT_EIG(
2,
complex, RKH)
250GET_ROOT_JEN(
2,
complex, RKS)
251GET_ROOT_JEN(
2,
complex, RKD)
252GET_ROOT_JEN(
2,
complex, RKH)
253GET_ROOT_LAG(
2,
complex, RKS)
254GET_ROOT_LAG(
2,
complex, RKD)
255GET_ROOT_LAG(
2,
complex, RKH)
256GET_ROOT_SKO(
2,
complex, RKD)
259 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
260 call disp%show(
"! Compute the roots of polynomials with complex coefficients with 12 zeros evenly distribute on a circle of radius 1. centered at 0+2i.")
261 call disp%show(
"!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
265[ (
4095._RKH,
0._RKH) \
266, (
0._RKH,
24576._RKH) \
267, (
-67584._RKH,
0._RKH) \
268, (
0._RKH,
-112640._RKH) \
269, (
126720._RKH,
0._RKH) \
270, (
0._RKH,
101376._RKH) \
271, (
-59136._RKH,
0._RKH) \
272, (
0._RKH,
-25344._RKH) \
273, (
7920._RKH,
0._RKH) \
274, (
0._RKH,
1760._RKH) \
275, (
-264._RKH,
0._RKH) \
276, (
0._RKH,
-24._RKH) \
279GET_ROOT_EIG(
2,
complex, RKS)
280GET_ROOT_EIG(
2,
complex, RKD)
281GET_ROOT_EIG(
2,
complex, RKH)
282GET_ROOT_JEN(
2,
complex, RKS)
283GET_ROOT_JEN(
2,
complex, RKD)
284GET_ROOT_JEN(
2,
complex, RKH)
285GET_ROOT_LAG(
2,
complex, RKS)
286GET_ROOT_LAG(
2,
complex, RKD)
287GET_ROOT_LAG(
2,
complex, RKH)
288GET_ROOT_SKO(
2,
complex, RKD)
Allocate or resize (shrink or expand) an input allocatable scalar string or array of rank 1....
This is a generic method of the derived type display_type with pass attribute.
This is a generic method of the derived type display_type with pass attribute.
Generate and return the value of the polynomial of arbitrary degree whose coefficients are specified ...
Generate and return the conversion of the input value to an output Fortran string,...
This module contains procedures and generic interfaces for resizing allocatable arrays of various typ...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
integer, parameter RKD
The double precision real kind in Fortran mode. On most platforms, this is an 64-bit real kind.
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
integer, parameter RKH
The scalar integer constant of intrinsic default kind, representing the highest-precision real kind t...
integer, parameter RKS
The single-precision real kind in Fortran mode. On most platforms, this is an 32-bit real kind.
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...
type(jenkins_type), parameter jenkins
This is a scalar parameter object of type jenkins_type that is exclusively used to signify the use of...
type(laguerre_type), parameter laguerre
This is a scalar parameter object of type laguerre_type that is exclusively used to signify the use o...
This module contains the generic procedures for converting values of different types and kinds to For...
Generate and return an object of type display_type.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example Windows Batch compile command via Intel ifort
compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example output ⛓
19+0.999999642,
+0.00000000,
20-0.429153442E-4,
+1.00021696,
21-0.429153442E-4,
-1.00021696,
22+0.427067280E-4,
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23+0.427067280E-4,
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25+0.114440918E-4,
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26+0.238418579E-5,
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27+0.238418579E-5,
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28+0.190734863E-5,
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29+0.190734863E-5,
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44+0.99999999999999922,
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45+0.95202123961968255E-9,
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47-0.95202196126464855E-9,
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48-0.95202196126464855E-9,
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72+0.108472542273474053061394483809842788E-16,
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79-0.308148791101957736488956470813588371E-32,
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223+1.00000000000000000000000000000000000,
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2201call cmplx_roots_gen(root, coef,
size(coef)
- 1,
.true.,
.true.)
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- Benchmarks:
Benchmark :: The effects of method
on runtime efficiency ⛓
- The following program compares the runtime performance of setPolyRoot using different polynomial root finding algorithms.
8 use iso_fortran_env,
only:
error_unit
12 integer(IK) ,
parameter :: degmax
= 9_IK
13 integer(IK) ,
allocatable :: rootCount(:)
17 integer(IK) :: fileUnit
18 integer(IK) :: rootUnit
19 real(RKG) :: dumsum
= 0._RKG
20 complex(RKG) :: coef(
0:
2**degmax)
21 complex(RKG) :: root(
2**degmax)
22 type(bench_type) ,
allocatable :: bench(:)
24 bench
= [
bench_type(name
= SK_
"Eigen", exec
= setPolyRootEigen, overhead
= setOverhead)
&
25 ,
bench_type(name
= SK_
"Jenkins", exec
= setPolyRootJenkins, overhead
= setOverhead)
&
26 ,
bench_type(name
= SK_
"Laguerre", exec
= setPolyRootLaguerre, overhead
= setOverhead)
&
28 ,
bench_type(name
= SK_
"Skowron", exec
= setPolyRootSkowron, overhead
= setOverhead)
&
31 allocate(rootCount(
size(bench)))
33 write(
*,
"(*(g0,:,' '))")
34 write(
*,
"(*(g0,:,' '))")
"Benchmarking setPolyRoot()"
35 write(
*,
"(*(g0,:,' '))")
37 open(newunit
= fileUnit, file
= "main.out", status
= "replace")
38 open(newunit
= rootUnit, file
= "root.count", status
= "replace")
40 write(fileUnit,
"(*(g0,:,','))")
"Polynomial Degree", (bench(ibench)
%name, ibench
= 1,
size(bench))
41 write(rootUnit,
"(*(g0,:,','))")
"Polynomial Degree", (bench(ibench)
%name, ibench
= 1,
size(bench))
42 loopOverArraySize:
do ideg
= 0, degmax
45 call setUnifRand(coef(
0 : degree), (
1._RKG,
1._RKG), (
2._RKG,
2._RKG))
46 write(
*,
"(*(g0,:,' '))")
"Benchmarking with coef size", degree
47 do ibench
= 1,
size(bench)
48 bench(ibench)
%timing
= bench(ibench)
%getTiming(minsec
= 0.07_RK)
50 write(rootUnit,
"(*(g0,:,','))") degree, rootCount
51 write(fileUnit,
"(*(g0,:,','))") degree, (bench(ibench)
%timing
%mean, ibench
= 1,
size(bench))
53 end do loopOverArraySize
54 write(
*,
"(*(g0,:,' '))") dumsum
55 write(
*,
"(*(g0,:,' '))")
65 subroutine setOverhead()
70 dumsum
= dumsum
+ rootCount(ibench)
73 subroutine setPolyRootEigen()
75 call setPolyRoot(root(
1 : degree), rootCount(ibench), coef(
0 : degree), method)
79 subroutine setPolyRootJenkins()
81 call setPolyRoot(root(
1 : degree), rootCount(ibench), coef(
0 : degree), method)
85 subroutine setPolyRootLaguerre()
87 call setPolyRoot(root(
1 : degree), rootCount(ibench), coef(
0 : degree), method)
92 subroutine setPolyRootSkowron()
94 call cmplx_roots_gen(root(
1 : degree), coef(
0 : degree), degree, polish_roots_after
= .true., use_roots_as_starting_points
= .false.)
95 rootCount(ibench)
= root(
1)
%re
Generate and return an object of type timing_type containing the benchmark timing information and sta...
Return a uniform random scalar or contiguous array of arbitrary rank of randomly uniformly distribute...
This module contains abstract interfaces and types that facilitate benchmarking of different procedur...
This module contains classes and procedures for computing various statistical quantities related to t...
integer, parameter RK
The default real kind in the ParaMonte library: real64 in Fortran, c_double in C-Fortran Interoperati...
This is the class for creating benchmark and performance-profiling objects.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Example Windows Batch compile command via Intel ifort
compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
Postprocessing of the benchmark output ⛓
3import matplotlib.pyplot
as plt
8dirname = os.path.basename(os.getcwd())
12df = pd.read_csv(
"main.out", delimiter =
",")
13colnames = list(df.columns.values)
19ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
22for colname
in colnames[1:]:
23 plt.plot( df[colnames[0]].values
28plt.xticks(fontsize = fontsize)
29plt.yticks(fontsize = fontsize)
30ax.set_xlabel(colnames[0], fontsize = fontsize)
31ax.set_ylabel(
"Runtime [ seconds ]", fontsize = fontsize)
32ax.set_title(
" vs. ".join(colnames[1:])+
"\nLower is better.", fontsize = fontsize)
36plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"-")
37ax.tick_params(axis =
"y", which =
"minor")
38ax.tick_params(axis =
"x", which =
"minor")
39ax.legend ( colnames[1:]
46plt.savefig(
"benchmark." + dirname +
".runtime.png")
52ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
55plt.plot( df[colnames[0]].values
56 , np.ones(len(df[colnames[0]].values))
61for colname
in colnames[2:]:
62 plt.plot( df[colnames[0]].values
63 , df[colname].values / df[colnames[1]].values
67plt.xticks(fontsize = fontsize)
68plt.yticks(fontsize = fontsize)
69ax.set_xlabel(colnames[0], fontsize = fontsize)
70ax.set_ylabel(
"Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
71ax.set_title(
"Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}.".format(colnames[1]), fontsize = fontsize)
75plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"-")
76ax.tick_params(axis =
"y", which =
"minor")
77ax.tick_params(axis =
"x", which =
"minor")
78ax.legend ( colnames[1:]
85plt.savefig(
"benchmark." + dirname +
".runtime.ratio.png")
91df = pd.read_csv(
"root.count", delimiter =
",")
92colnames = list(df.columns.values)
94ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
97plt.plot( range(1, df[colnames[0]].values[-1] * 2)
98 , range(1, df[colnames[0]].values[-1] * 2)
103for colname
in colnames[1:-1]:
104 plt.plot( df[colnames[0]].values
109plt.xticks(fontsize = fontsize)
110plt.yticks(fontsize = fontsize)
111ax.set_xlabel(colnames[0], fontsize = fontsize)
112ax.set_ylabel(
"Number of Roots Found", fontsize = fontsize)
113ax.set_title(
" vs. ".join(colnames[1:])+
"\nHigher is better.", fontsize = fontsize)
117plt.grid(visible =
True, which =
"both", axis =
"both", color =
"0.85", linestyle =
"-")
118ax.tick_params(axis =
"y", which =
"minor")
119ax.tick_params(axis =
"x", which =
"minor")
120ax.legend ( [
"Equality"] + list(colnames[1:])
123 , fontsize = fontsize
127plt.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
- Todo:
- Critical Priority: The current implementation relies on several local allocations, the most important of which are called
workspace
in the current implementation.
Although this generic interface used to accept a workspace
argument, it was subsequently removed in favor of simplicity.
A new set of interfaces can be added in the future to allow specification of scratch space arguments to avoid repeated local allocations.
This could boost performance when this generic interface is called many times repeatedly.
- Todo:
- Critical Priority: The generic interface for the Eigenvalue method internally uses the eigenvalue computing routine of EISPACK for upper Hessenberg matrices to compute the roots of the polynomial.
This internal implementation should be substituted with the corresponding external routine once it is implemented in the ParaMonte library.
- Todo:
- Low Priority: The existing implementations of the algorithms for the Jenkins-Traub method contain relics of F77 coding style in the form of a few
goto
statements.
These remaining goto statements should be carefully removed in the future.
- Todo:
- Critical Priority: The method of Skowron-Gould must be fully implemented.
- Todo:
- Normal Priority: A benchmark comparing the runtime performances of the
complex
vs. real
coefficient routines of this module should be added here.
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
Copyright © 1996 California Institute of Technology, Pasadena, California. ALL RIGHTS RESERVED. Based on Government Sponsored Research NAS7-03001. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: Redistributions of source code must retain this copyright notice, this list of conditions and the following disclaimer. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. Neither the name of the California Institute of Technology (Caltech) nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. For those codes indicated with a Math a la Carte copyright, the same rules apply, except without the full force of the Caltech legal team. When citing this software we request that you also mention the names of the people who wrote the software you are using. Designed by C. L. Lawson, JPL, May 1986. Programmed by C. L. Lawson and S. Y. Chiu, JPL, May 1986, Feb. 1987.
- Author:
- Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan
Definition at line 4608 of file pm_polynomial.F90.