Any availiable function to find the Taylor series of multi variables function in Matlab?

I am looking for a ready to use function in Matlab to find the Taylor expansion of a functions with two or more variables (e.g. tanh(x^7*y^9^z^5)). Please let me know if there is such function in Matlab.

Answer 1

[Torsten]

https://uk.mathworks.com/matlabcentral/answers/92764-how-do-i-generate-the-taylor-series-as-a-function-of-several-variables-in-matlab-r2011b

syms x y z
f(x,y,z) = tanh(x^7*y^9*z^5);
taylor(f,[x y z],[1 1 1],'Order',2)

ans(x, y, z) = 

[Mehdi]

what is problem with my function where I receive 1 instead of taylor expansion?

syms xx yy
k=100;
f=- vpa(2146702909675882809503682033933399905335826325*xx^9*yy^9)/11150372599265311570767859136324180752990208 + (1587967252519403636411870604735180043125989625*xx^9*yy^8)/5575186299632655785383929568162090376495104 + (17639360745426635511855086638766468926126459875*xx^9*yy^7)/44601490397061246283071436545296723011960832 - (431284328058774504067793959976795724976545555*xx^9*yy^6)/696898287454081973172991196020261297061888 - (12993287722661922638788467553649639108437064835*xx^9*yy^5)/44601490397061246283071436545296723011960832 + (1206817075246069632318716986669541278160772775*xx^9*yy^4)/2787593149816327892691964784081045188247552 + (5138909461003175489938484170634052266819688725*xx^9*yy^3)/44601490397061246283071436545296723011960832 - (72716798311978341010558827315982986191821905*xx^9*yy^2)/696898287454081973172991196020261297061888 - (1197236208181378637639504269592639035279087665*xx^9*yy)/44601490397061246283071436545296723011960832 + (30423874459994412977383604476886160940746185*xx^9)/5575186299632655785383929568162090376495104 + (8094790880015327525694605814920739418439287725*xx^8*yy^9)/2787593149816327892691964784081045188247552 - (285743684916570536194588196441080828723328178675*xx^8*yy^8)/89202980794122492566142873090593446023921664 - (6686861200533386632065997818427854246215113305*xx^8*yy^7)/696898287454081973172991196020261297061888 + (157001869330425518481531763580902779395436599415*xx^8*yy^6)/22300745198530623141535718272648361505980416 + (15350689937843699961175740256400109996121380375*xx^8*yy^5)/1393796574908163946345982392040522594123776 - (206512033439850904054937113093163624192322042825*xx^8*yy^4)/44601490397061246283071436545296723011960832 - (3479476522267890993628796487849129439635143625*xx^8*yy^3)/696898287454081973172991196020261297061888 + (18712604797880071317805036942199122521197359575*xx^8*yy^2)/22300745198530623141535718272648361505980416 + (1869246621670048362557342074310025153518449965*xx^8*yy)/2787593149816327892691964784081045188247552 - (4089215965643055747590786827106386135115380275*xx^8)/89202980794122492566142873090593446023921664 + (216255546256559295251079313253452049445763455*xx^7*yy^9)/348449143727040986586495598010130648530944 - (868641325364973493898126340263842300348545855*xx^7*yy^8)/1393796574908163946345982392040522594123776 - (14785537121406447202257499440081382142298519099*xx^7*yy^7)/11150372599265311570767859136324180752990208 + (116540829629507365267125159526451609264014215*xx^7*yy^6)/87112285931760246646623899502532662132736 + (11170081785792631086653879206603595320491089331*xx^7*yy^5)/11150372599265311570767859136324180752990208 - (652299342907430898149182084981866414949696905*xx^7*yy^4)/696898287454081973172991196020261297061888 - (4303517165264733669855129139552505045324631645*xx^7*yy^3)/11150372599265311570767859136324180752990208 + (10578825782023300845453772557509072093336001*xx^7*yy^2)/43556142965880123323311949751266331066368 + (998213736763384913910074759047227544847506773*xx^7*yy)/11150372599265311570767859136324180752990208 - (29946355461657315300256240552185966952551471*xx^7)/1393796574908163946345982392040522594123776 - (1277356081222180962342283013232991241852904465*xx^6*yy^9)/696898287454081973172991196020261297061888 + (57447439083834576362467553225131370438848237035*xx^6*yy^8)/22300745198530623141535718272648361505980416 + (1173296429365947392287371443632107462978009165*xx^6*yy^7)/174224571863520493293247799005065324265472 - (23673134207774883972271882396704370580007933039*xx^6*yy^6)/5575186299632655785383929568162090376495104 - (2937701213452088192123555543440803264914467299*xx^6*yy^5)/348449143727040986586495598010130648530944 + (14159347676475748959036290080103848146860867025*xx^6*yy^4)/11150372599265311570767859136324180752990208 + (749877940244270735637721966049124917356845885*xx^6*yy^3)/174224571863520493293247799005065324265472 + (782685832362921584689673760969891945953777553*xx^6*yy^2)/5575186299632655785383929568162090376495104 - (539977758872163289054492124375185771143918033*xx^6*yy)/696898287454081973172991196020261297061888 + (5795161625895678368156852916105373987594511979*xx^6)/22300745198530623141535718272648361505980416 - (5023626067733175609651265492842895195168362165*xx^5*yy^9)/5575186299632655785383929568162090376495104 + (2207379816207475241162406248223006569040862935*xx^5*yy^8)/2787593149816327892691964784081045188247552 + (47950825635610780986659544491454706340397108297*xx^5*yy^7)/22300745198530623141535718272648361505980416 - (588774433706353379897742534304221654039246663*xx^5*yy^6)/348449143727040986586495598010130648530944 - (44608078263668464626393951292252447406629869273*xx^5*yy^5)/22300745198530623141535718272648361505980416 + (1613038118657167505912389296857854524947676825*xx^5*yy^4)/1393796574908163946345982392040522594123776 + (23570688854853763073042723518782612790921757535*xx^5*yy^3)/22300745198530623141535718272648361505980416 - (113510140727511300460098712979462156361337425*xx^5*yy^2)/348449143727040986586495598010130648530944 - (6817973449093402642853212701104432585928821163*xx^5*yy)/22300745198530623141535718272648361505980416 + (184838927094446995029201369223921105703104647*xx^5)/2787593149816327892691964784081045188247552 + (31380186488931551370058361496245928395816772575*xx^4*yy^9)/1393796574908163946345982392040522594123776 - (1758702445038817232726176779731884586549332868025*xx^4*yy^8)/44601490397061246283071436545296723011960832 - (33218490572036542393092937176469859040906121155*xx^4*yy^7)/348449143727040986586495598010130648530944 + (843981485493394825713526892530506348990296828805*xx^4*yy^6)/11150372599265311570767859136324180752990208 + (94251624724512021502035994822030873708141367565*xx^4*yy^5)/696898287454081973172991196020261297061888 - (765302392604646459013613426858243443467023490875*xx^4*yy^4)/22300745198530623141535718272648361505980416 - (26051472095770585704126329008135447818638784275*xx^4*yy^3)/348449143727040986586495598010130648530944 - (9551461763890264957289963973620923748598225435*xx^4*yy^2)/11150372599265311570767859136324180752990208 + (17196469545705046799299985950707233685621881055*xx^4*yy)/1393796574908163946345982392040522594123776 - (48412290717709997717153300332089796247538326265*xx^4)/44601490397061246283071436545296723011960832 + (941109349474535911451616661821106567867537125*xx^3*yy^9)/1393796574908163946345982392040522594123776 - (1174244552874873223035231031480900497934023075*xx^3*yy^8)/1393796574908163946345982392040522594123776 - (20361225581568567923686744589522827658576624955*xx^3*yy^7)/11150372599265311570767859136324180752990208 + (39584968580329795728950940517214770307434335*xx^3*yy^6)/21778071482940061661655974875633165533184 + (23458516464006675395891679247259419002768896835*xx^3*yy^5)/11150372599265311570767859136324180752990208 - (851688199122087410134053760306093104684621525*xx^3*yy^4)/696898287454081973172991196020261297061888 - (15637727799880882327290754576104647826715168925*xx^3*yy^3)/11150372599265311570767859136324180752990208 + (29341459645317546529685572705520876577051855*xx^3*yy^2)/87112285931760246646623899502532662132736 + (5011420945327438626354964312196465908094234685*xx^3*yy)/11150372599265311570767859136324180752990208 - (125283292999146417157156696376640452081866835*xx^3)/1393796574908163946345982392040522594123776 - (35643509355104072817665294345590475660747146425*xx^2*yy^9)/696898287454081973172991196020261297061888 + (1872760743346397986120124413411813119412045269675*xx^2*yy^8)/22300745198530623141535718272648361505980416 + (36390552938954376406834468187448925576623439893*xx^2*yy^7)/174224571863520493293247799005065324265472 - (930314746723434588666177195703059675161177190255*xx^2*yy^6)/5575186299632655785383929568162090376495104 - (100809382380090436397261413740272360141145204891*xx^2*yy^5)/348449143727040986586495598010130648530944 + (929769947314964740179937673332890647768037984465*xx^2*yy^4)/11150372599265311570767859136324180752990208 + (27287439738914744607616926917914225474665410565*xx^2*yy^3)/174224571863520493293247799005065324265472 - (11540959773500599403794316292492996114189538863*xx^2*yy^2)/5575186299632655785383929568162090376495104 - (17449701902039745490242163912540688306429882361*xx^2*yy)/696898287454081973172991196020261297061888 + (35122173917479363738100862234581108137514304171*xx^2)/22300745198530623141535718272648361505980416 - (2255097230860381206152749351617455809672044745*xx*yy^9)/11150372599265311570767859136324180752990208 + (2168816628024980374461014350770096009019357665*xx*yy^8)/5575186299632655785383929568162090376495104 + (27046038795224386955728969793334632924015008227*xx*yy^7)/44601490397061246283071436545296723011960832 - (590212436135125327923049635849260481403670583*xx*yy^6)/696898287454081973172991196020261297061888 - (36304948749180317956941914133403396762716230691*xx*yy^5)/44601490397061246283071436545296723011960832 + (1583056855557692418384969876461998197073089695*xx*yy^4)/2787593149816327892691964784081045188247552 + (27484692689867334306687311759874973819976026005*xx*yy^3)/44601490397061246283071436545296723011960832 - (104255809907916433055923335622932126645726549*xx*yy^2)/696898287454081973172991196020261297061888 - (9205355621994819342146712860571987786619361601*xx*yy)/44601490397061246283071436545296723011960832 + (220816865194317615868568855814620996552449073*xx)/5575186299632655785383929568162090376495104 + (76828297887427851822683521168415270943435162685*yy^9)/2787593149816327892691964784081045188247552 - (3917684154726736823398471536296978037714283086195*yy^8)/89202980794122492566142873090593446023921664 - (77131555128675321096947207038878222843991869993*yy^7)/696898287454081973172991196020261297061888 + (1970986683407627074325019523003479974617451789943*yy^6)/22300745198530623141535718272648361505980416 + (211134394987302797546644924545169826774270265159*yy^5)/1393796574908163946345982392040522594123776 - (2038600361316622246653155899145012259420048867785*yy^4)/44601490397061246283071436545296723011960832 - (56566850002827011453690682806041619180254985625*yy^3)/696898287454081973172991196020261297061888 + (43423414494451507811145033075147441881593811799*yy^2)/22300745198530623141535718272648361505980416 + (35696532930567486560276536615522532283474689213*yy)/2787593149816327892691964784081045188247552 - 62755544772437504320590342390381422715234113715/89202980794122492566142873090593446023921664;
g=0.5*(1+tanh(k*f));
vpa(taylor(g,[xx yy],[.1 .1],'Order',8))

ans = 

1.0

[Torsten]

From the result I would conclude that the derivatives up to order 8 are (numerically) 0 at [0.1,0.1].

[Mehdi]

what do you recommend to get a good estimation of my function in -1<x,y<1?

[Walter Roberson]

syms xx yy
k=100;
f=- str2sym('(2146702909675882809503682033933399905335826325*xx^9*yy^9)/11150372599265311570767859136324180752990208 + (1587967252519403636411870604735180043125989625*xx^9*yy^8)/5575186299632655785383929568162090376495104 + (17639360745426635511855086638766468926126459875*xx^9*yy^7)/44601490397061246283071436545296723011960832 - (431284328058774504067793959976795724976545555*xx^9*yy^6)/696898287454081973172991196020261297061888 - (12993287722661922638788467553649639108437064835*xx^9*yy^5)/44601490397061246283071436545296723011960832 + (1206817075246069632318716986669541278160772775*xx^9*yy^4)/2787593149816327892691964784081045188247552 + (5138909461003175489938484170634052266819688725*xx^9*yy^3)/44601490397061246283071436545296723011960832 - (72716798311978341010558827315982986191821905*xx^9*yy^2)/696898287454081973172991196020261297061888 - (1197236208181378637639504269592639035279087665*xx^9*yy)/44601490397061246283071436545296723011960832 + (30423874459994412977383604476886160940746185*xx^9)/5575186299632655785383929568162090376495104 + (8094790880015327525694605814920739418439287725*xx^8*yy^9)/2787593149816327892691964784081045188247552 - (285743684916570536194588196441080828723328178675*xx^8*yy^8)/89202980794122492566142873090593446023921664 - (6686861200533386632065997818427854246215113305*xx^8*yy^7)/696898287454081973172991196020261297061888 + (157001869330425518481531763580902779395436599415*xx^8*yy^6)/22300745198530623141535718272648361505980416 + (15350689937843699961175740256400109996121380375*xx^8*yy^5)/1393796574908163946345982392040522594123776 - (206512033439850904054937113093163624192322042825*xx^8*yy^4)/44601490397061246283071436545296723011960832 - (3479476522267890993628796487849129439635143625*xx^8*yy^3)/696898287454081973172991196020261297061888 + (18712604797880071317805036942199122521197359575*xx^8*yy^2)/22300745198530623141535718272648361505980416 + (1869246621670048362557342074310025153518449965*xx^8*yy)/2787593149816327892691964784081045188247552 - (4089215965643055747590786827106386135115380275*xx^8)/89202980794122492566142873090593446023921664 + (216255546256559295251079313253452049445763455*xx^7*yy^9)/348449143727040986586495598010130648530944 - (868641325364973493898126340263842300348545855*xx^7*yy^8)/1393796574908163946345982392040522594123776 - (14785537121406447202257499440081382142298519099*xx^7*yy^7)/11150372599265311570767859136324180752990208 + (116540829629507365267125159526451609264014215*xx^7*yy^6)/87112285931760246646623899502532662132736 + (11170081785792631086653879206603595320491089331*xx^7*yy^5)/11150372599265311570767859136324180752990208 - (652299342907430898149182084981866414949696905*xx^7*yy^4)/696898287454081973172991196020261297061888 - (4303517165264733669855129139552505045324631645*xx^7*yy^3)/11150372599265311570767859136324180752990208 + (10578825782023300845453772557509072093336001*xx^7*yy^2)/43556142965880123323311949751266331066368 + (998213736763384913910074759047227544847506773*xx^7*yy)/11150372599265311570767859136324180752990208 - (29946355461657315300256240552185966952551471*xx^7)/1393796574908163946345982392040522594123776 - (1277356081222180962342283013232991241852904465*xx^6*yy^9)/696898287454081973172991196020261297061888 + (57447439083834576362467553225131370438848237035*xx^6*yy^8)/22300745198530623141535718272648361505980416 + (1173296429365947392287371443632107462978009165*xx^6*yy^7)/174224571863520493293247799005065324265472 - 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62755544772437504320590342390381422715234113715/89202980794122492566142873090593446023921664')

f = 

g=0.5*(1+tanh(k*f));
fsurf(g, [-1 1 -1 1])

You have steep boundaries in one direction. You are not going to be able to get a decent approximation without using a lot of terms.

[Mehdi]

suppose f=x. Still could not estimate f even with a lot of terms.

syms x
k=10;
f=x;
g=0.5*(1+tanh(k*f));
fplot(g,[-1,1])

fplot((taylor(g,[x],[0],'Order',88)),[-1,1])

d

[Torsten]

what do you recommend to get a good estimation of my function in -1<x,y<1?

Why not just evaluating the function ?

[Mehdi]

I need to estimate this function (g) with polynomials so the Taylor series expansion is the best choice, but unsuccessful yet.

[Torsten]

suppose f=x. Still could not estimate f even with a lot of terms.

The Taylor series for tanh(x) converges for |x| < pi/2, and you will need many terms to make it converge when you approach the boundaries.

If you change your command to

fplot((taylor(g,[x],[0],'Order',88)),[-0.15,0.15])

you will see that the behaviour of the Taylor approximation near 0 is correct.

Answer 2

[the cyclist]

syms x y z;
f = tanh(x^7*y^9*z^5) % I think you may have had a typo or two in your function, so check this

f = 

taylor_series = taylor(f, [x, y, z], 'Order', 127);
disp(taylor_series);