<content> <p>PartialFrac</p> </content>

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Partial fractions decomposition

f# x# k# Coeff #VAR# 0 : diff(equrep(f# num2str, x# num2str,#VAR#),#VAR#, k# num2str) concat str2num k#! / 21 line :

f# x# Coeff C# 0 : k# MaxDegree 0 range d# f# x# k# Coeff : num2str 0 ≠ C# k# 1 + el d# : if for C# 31 line :

x# tol# ZRound n# 15 tol# log10 trunc abs 12 mat min eval : t# x# eval : 12 mat t# t# Re t# Re abs 1000 tol# * > (* vectorize i t# Im t# Im abs 1000 tol# * > (* vectorize * + eval : t# Re n# round vectorize i t# Im n# round vectorize * + eval 31 line :

C# x# Poly s# C# 1 el : k# 2 : k# C# length ≤ k# k# 1 + : s# s# C# k# el x# k# 1 -^* + : for s# 31 line :

OPolyRoots try to improve accuracy of polyroots, returning also the multiplicity of the roots. C# OPolyRoots C# length 2 < 00013 mat R# C# 1 C# length range el polyroots eval TOL ZRound : n# R# length : r# 1 n# range : D# 0 : Q# 0 : O# 0 : 15 mat D# r# el r# 1 + (! (C# r# 1 + el * : Q# r# el R# 1 r# range el R# r# el 1 findrows length eval : O# r# el R# R# r# el 1 findrows length eval : k# 1 n# range t# R# k# el : O# k# el 1 ≡ (Q# k# el 1 ≡ (& iter# 110 range u# t# : t# u# C# u# Poly D# u# Poly TOL + / - eval : u# t# - abs 0.01 TOL * < break if 31 line for Q# k# el 1 > R# k# el R# k# 1 - el : if if 21 line for R# Q# O# 13 mat 71 line if 11 line :

OPolyRoots try to improve accuracy of polyroots. C# PolyRoots C# OPolyRoots 1 el 11 line :

Partial fraction decomposition. f# x# ParFrac A# B# 12 mat f# numden 1 el x# Coeff f# numden 2 el x# Coeff 12 mat : K# 011 mat : na# A# length : nb# B# length : 13 mat n# nb# 1 - : r# 1 n# range : c# r# : M# 0 : xo# 0 : 15 mat na# nb# ≥ h# na# nb# - (0 range K# h# 1 + el A# nb# h# + el B# nb# el / : k# 1 nb# range A# h# k# + el A# h# k# + el B# k# el K# h# 1 + el * - : for 21 line for A# A# r# el : 21 line if R# Q# O# 13 mat B# OPolyRoots : n# 1 ≡ K# A# B# nb# el / R# Q# O# augment 12 mat xo# r# el R# r# el TOL + R# 2 n# range el R# 2 n# range 1 - el - norme n# 1 + / r# * + eval : a# r# el A# xo# r# el Poly B# xo# r# el Poly / eval : x# Co# #VAR# k# num2str concat str2num x# R# k# el - (Q# k# el^/ k# 1 n# sum 11 line : M# r# c# el xo# r# el Co# #VAR# c# num2str concat str2num diff : K# M# 1 -^a# * 0.001 TOL * ZRound R# Q# O# augment 12 mat 51 line if 61 line :

Integrate the rational function f by decomposing it in partial fractions. f# x# a# b# IParFrac K# F# 12 mat f# x# ParFrac : u# k# I# u# F# k# 2 el - ln F# k# 3 el 1 ≡ u# F# k# 2 el - (1 F# k# 3 el -^1 F# k# 3 el - / cases : K# k# el b# k#^a# k#^- k# / * k# 1 K# rows sum F# k# 1 el b# k# I# a# k# I# - (* k# 1 F# rows sum + TOL ZRound 31 line :

Utility for show the partial fraction decomposition f# x# pf K# F# 12 mat f# x# ParFrac eval : K# k# 1 + el x# k#^* k# 0 K# rows 1 - sum F# k# 1 el x# F# k# 2 el - (F# k# 3 el^/ k# 1 F# rows sum + 21 line :

MaxDegree 20 :

TOL 1010 -^:

K F 12 mat f x ParFrac ≡

Returns a vector K and a matrix F with the partial fractions decomposition of a quotient of polynomials f in the variable x. The rational function f could be restored with pf(f,x)

f a.0 a.1 x * + a.2 x 2^* + \002E\\002E\\002E\ + b.0 b.1 x * + b.2 x 2^* + \002E\\002E\\002E\ + / ≡

f x ParFrac K.0 K.1 \002E\\002E\\002E\ 31 mat A.1 r.1 q.1 o.1 A.2 r.2 q.2 o.2 \002E\\002E\\002E\ \0020\ \0020\ \0020\ 34 mat 12 mat ≡

Also

f x pf K.0 K.1 x * + K.2 x 2^* + \002E\\002E\\002E\ + A.1 x r.1 - (q.1^/ + A.2 x r.2 - (q.2^/ + \002E\\002E\\002E\ + ≡

f x x.1 x.2 IParFrac f x x.1 x.2 int ≡

References

https://www.mathworks.com/help/matlab/ref/residue.html

https://lpsa.swarthmore.edu/BackGround/PartialFraction/PartialFraction.html

Examples

f 6 s 2^* 3 s * + 2 + 2 s 2^* 14 s * + 20 + / :

f s ParFrac 311 mat 3.3333 2 - 11 22.8333 - 5 - 1124 mat 12 mat

f s pf 3 s 0^* 3.333333333333 s 2 - (- (1^/ 22.833333333333 - s 5 - (- (1^/ + +

f s 23 IParFrac 0.694845039

f s 23 int 0.694845039454177

f 4 s * - 8 + s 2^6 s * + 8 + / :

f s ParFrac 011 mat 82 - 1112 - 4 - 1124 mat 12 mat

f s pf 0 s 0^* 8 s 2 - (- (1^/ 12 - s 4 - (- (1^/ + +

f s 23 IParFrac 0.064659747 -

f s 23 int 0.064659747413126 -

f 2 s 4^* s + s 2^1 + / :

f s ParFrac 2 - 0231 mat 0.5 i - i 11 0.5 i + i - 1124 mat 12 mat

f s pf 2 - (s 0^* 0 s 1^* + 2 s 2^* + 0.5 i - s i - (1^/ 0.5 i + s i - (- (1^/ + +

f s 23 IParFrac 11.297034366

f s 23 int 11.2970343661559

f 2 s 3^* s 2^+ s 3^s + 1 + / :

f s ParFrac 211 mat 0.070835812033 - 0.682327804 - 11 0.535417905934 1.038991708586 i * + 0.341163902 1.1615414 i * + 11 0.535417905973 1.038991708595 i * - 0.341163902 1.1615414 i * - 1134 mat 12 mat

IParFrac returns the values of this integral

F ω 110114 mat PolyRoots : 2 s * ω k el 2^2 ω k el * - 2 - (s ω k el - ln * 3 ω k el 2^* 1 + / (k 13 sum + 21 line :

f s 23 IParFrac 1.94913378

f s 23 int 1.94913378036444

F 23 at 1.94913378033686

f s 3 + s 3^7 s 2^* + 10 s * + / :

f s ParFrac 011 mat 0.3 011 0.1667 - 2 - 11 0.1333 - 5 - 1134 mat 12 mat

f s pf 0 s 0^* 0.3 s 0 - (1^/ 0.166666666667 - s 2 - (- (1^/ + 0.133333333333 - s 5 - (- (1^/ + +

f s 3 + s 5 + (s 2^4 s * + 5 + (* / :

f s ParFrac 011 mat 0.1 0.2 i * - 2 - i + 11 0.1 0.2 i * + 2 - i - 11 0.2 - 5 - 1134 mat 12 mat

f s pf 0 s 0^* 0.1 0.2 i * - s 2 - i + (- (1^/ 0.1 0.2 i * + s 2 - i - (- (1^/ + 0.2 - s 5 - (- (1^/ + +

Usually, for the cases of repeated poles, we need to increse 'TOL

TOL 11010 -^*

y x 3 + x x 2 + (2^* x 5 + (* / :

y x ParFrac 011 mat 0.15 011 8.6805565 105^* - 1.9999999 - 11 8.6805545 105^* 2.0000001 - 11 0.0444444 5 - 1144 mat 12 mat

TOL 107 -^:

y x ParFrac 011 mat 0.15 011 0.1944 - 2 - 12 0.1667 - 2 - 22 0.0444 5 - 1144 mat 12 mat

Tip: increase 'TOL until ParFrac recognize repeated poles.

y x pf 0 x 0^* 0.15 x 0 - (1^/ 0.194444444 - x 2 - (- (1^/ + 0.166666667 - x 2 - (- (2^/ + 0.044444444 x 5 - (- (1^/ + +

y x 23 IParFrac 0.015032

y x 23 int 0.015032137577468

TOL 104 -^:

y 0.1 x 8^* 2 x * - 3 + 2 x 2^* x 4 - (3^* x 6 - (2^* / :

y x ParFrac 1.2 0.05 21 mat 0.000271 - 012 0.000651 - 022 163.175098 413 127.964844 423 51.160939 433 145.774826 - 612 291.584375 62274 mat 12 mat

yp y x pf :

yp 65 / x 0^* 120 / x 1^* + 2711000000 / - x 0 - (1^/ 6511000000 / - x 0 - (2^/ + 81587549500000 / x 4 - (1^/ + 31991211250000 / x 4 - (2^/ + 511609391000000 / x 4 - (3^/ + 72887413500000 / - x 6 - (1^/ + 93307320 / x 6 - (2^/ + +

y x 23 IParFrac 0.747 -

y 0.1 + yp 0.1 - 2 1 sys

y x 23 int 0.747004981643018 -

Alvaro

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