Di Nardo, Elvira

Guarino, G.

Senato, D.

English

DI NARDO, Elvira

Institutional Research Information System University of Turin

1 - 15This Maple worksheet accompanies the paper [1]: Di Nardo E., G. Guarino, D. Senato (2011), A new algorithm for computing the multivariate Faà di Bruno’s formula, Applied Mathematics and Computation. doi:10.1016/j.amc.2011.01.001 A new algorithm for computing the multivariate Faà di Bruno’s formulaE. Di Nardo*elvira.dinardo@unibas.ithttp://www.unibas.it/utenti/dinardo/home.html;Tel: +39 0971205890, Fax: +39 0971205896G. Guarino**giuseppe.guarino@aspbasilicata.itD. Senato*domenico.senato@unibas.it  * Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, viale dell'Ateneo Lucano n.10, 85100 Potenza, Italy**Medical School, Università del Sacro Cuore (Rome branch), Largo Agostino Gemelli n.8, 00168 Roma, Italy2 - 15> > > (2.1)> > IntroductionAbstract: We provide a new algorithm for computing the multivariate Faà di Bruno's formula. We follow a symbolic approach based on the classical umbral calculus that leads back the computation ofthe multivariate Faà di Bruno's formula to a suitable multinomial expansion. The resulting computational times are faster compared with procedures existing in the literature.Application Areas/Subject: Combinatorics & algebraic methodsKeyword: multivariate composite function, Faà di Bruno's formula, multivariate cumulants, multivariate Hermite polynomials, classical umbral calculus.See Also:  background on the classical umbral calculus in [2], background on multiset subdivisions in[3], for different applications of multiset subdivisions see [5]  Initializationrestart;with combinat, partition, multinomial, permute ;with ListTools, Transpose, Flatten ;partition, multinomial, permuteTranspose, FlattenSubdivisions of a multisetThis topic has been extensively discussed in [3]. See http://www.maplesoft.com/applications/view.aspx?SID=33039 for details.nRep d proc u  mul x2!, x = convert u, multiset ; end:URv d proc u, v   local U, ou, i, ptr, vI;   oudNULL; Ud ; vId indets v ;   for ptr from nops u  by K1 to 2 do       if has uptr, v  then break; fi; od;   for i from ptr to nops u  do       if not ui = ou or has ui, vI  then           oud ui;          Ud op U ,  op u1 ..iK1 , ui$v, op uiC1 ..K1 ;       fi;       od; op U , op u , v ; end:URV d proc    local U, V, i;    Ud args1, 1 ; Vd args2, 1;    for i from 1 to nops V  do3 - 15(3.1)> > > > (3.2)>     U d  seq  URv u, Vi , u = U  ;     od; seq x, args1, 2$args2, 2 , x = U  ; end:URmV d proc   local U, i;   if nargs = 1 then       args;   else UdURV  args1, args2 ;        for i from 3 to nargs do            Ud seq  URV  u, argsi , u = U  ;        od; seq x1,x2nRep x1, x = U ;     fi; end:comb d proc V, ptr, Y    if ptr = nops V C1 then return Y ; fi;    seq  comb V, ptrC1,  op Y , L  , L = Vptr ;   end:makeTab d proc     local U;   Ud seq `if` argsi = 0, NULL,              seq seq aiz, z = y , multinomial argsi, seq r, r = y ,            y = partition argsi  ,        i = 1 ..nargs ;   if nops U = 1       then seq x1,x2nRep x1, x = op U ;      else  seq URmV op x , x = comb U, 1, ; fi; end:ExamplesmakeTab 3 ;a1, a1, a1 , 1 , a1, a12 , 3 , a13 , 1makeTab 2, 1 ;a1 a2, a1 , 2 , a1, a1, a2 , 1 , a12 a2 , 1 , a12, a2 , 1Multivariate Faà di Bruno s formula4 - 15A MAPLE algorithm for the computation of the multivariate Faà di Bruno’s formula by using the umbral equivalence (18) in [1].Univariate and Multivariate CaseMFB FunctionsIntroductionAs we have proved in [3], the function makeTab generates a multiset subdivision (see the output of the function makeTab in the previous section).  Parameters: MFB n1, n2 ... , ni 2;Univariate with UnivariateBy using the subdivisions of a multiset with all equal elements, we are able to compute theunivariate Faà di Bruno's formula, that is to expand dndxn f g x . For example in order to expand d2dx2 f g x  we call MFB(2) which gives  f2 g12Cf1 g2 as output . To see that f2 g12Cf1 g2 is the expansion of d2dx2 f g x , replace fi = Di f g x  for i = 1, 2 and  g1 = ddx g x , g2 =d2dx2 g x . Then we have D 2 f g x  ddx g x2CD f g x  d2dx2 g x .Remark: the function MFB uses the function makeTab as we show in the following example. Calling makeTab(2) we have a1, a1 , 1 , a12, 1 . The first block a1, a1  has two elements: its cardinality gives the index of  f2. The second block a12  has one element: its cardinality gives the index of f1. The first element a1 in the first block a1, a1  has degree 1: its degree gives the index of g1. Since in a1, a1   we have two equal elements of degree 1, we getg12. This term must be multiplied with the integer (the multiplicity) that appears in each block (for example: 1 in a1, a1 , 1  or n in a1,....., a1 , n ). The second block a12, 1  contains only one element with degree 2. This gives just g2. At the end, the summation of the two terms returns f2 g12Cf1 g2. Univariate with MultivariateWhen in the multiset there are two or more different elements, the result of a call to MFB is different from the previous one. For esample MFB(1,1) generates the following output: f1 g1, 1Cf2 g1, 0 g0, 1 that represents the expansion of 5 - 15> (4.1.3.1)> > (4.1.3.2)v2vx2 vx1 f g x1, x2 . Indeed if in f1 g1, 1C f2 g1, 0 g0, 1we replace fi = Di f g x1, x2  for i = 1, 2,g1, 1 = v2vx1 vx2 g x1, x2  and g1, 0 = vvx1 g x1, x2 , g0, 1 = vvx2 g x1, x2  we haveD 2 f g x1, x2  vvx2 g x1, x2  vvx1 g x1, x2 CD f g x1,x2  v2vx1 vx2 g x1, x2 , which is the well known expansion of  v2vx1 vx2 f g x1, x2Remark: Similarly to what it happens for the univariate/univariate case,  the function MFB calls the function makeTab as we show in the following example. Calling makeTab(1,1) we obtain a1 a2 , 1 , a1, a2 , 1 . The first block a1 a2  has one element: its cardinality gives the index of  f1. The second block a1, a2  has two elements: its cardinality gives the index of f2. In the blocks, two variables a1,a2 are involved. This explains why we use two indexes in  gi, j elements. In particular we have:  a1a2/g1, 1, a1/g1, 0 and  a2/g0, 1. Therefore from the block  a1 a2  we have g1, 1, and from the block a1, a2  we get g1, 0 g0, 1. At the end, the summation of the two terms returns f1 g1, 1Cf2 g1, 0 g0, 1. The CodeMFB :=procoption remember;local n, vIndets, E;n := add argsi, i = 1 ..nargs ;if n = 0 then return 1 end if;vIndets := seq ai, i = 1 ..nargs ;E := add fnops y1$y 2 $mul gseq degree x, vIndetsi, i = 1 .. nops vIndets , x = y1 , y= makeTab argsend proc:ExamplesUnivariate with Univariate dndxn f g xMFB 2 ;f2 g12Cf1 g2MFB 3 ;6 - 15(4.1.3.4)> (4.1.3.2)> > > (4.1.3.6)(4.1.3.3)(4.1.3.5)> f3 g13C3 f2 g1 g2C f1 g3Univariate with Multivariate vnvx1 ..vxn f g x1,.., xnMFB 1, 0 ;f1 g1, 0MFB 1, 1 ;f1 g1, 1Cf2 g1, 0 g0, 1MFB 2, 1 ;2 f2 g1, 1 g1, 0Cf3 g1, 02  g0, 1Cf1 g2, 1Cf2 g2, 0 g0, 1MFB 1, 1, 1 ;f1 g1, 1, 1C f2 g1, 1, 0 g0, 0, 1C f2 g1, 0, 1 g0, 1, 0C f2 g1, 0, 0 g0, 1, 1C f3 g1, 0, 0 g0, 1, 0 g0, 0, 1The function jointIntroductionThe procedure joint provides a way to multiply the factors of formula (22) in [1].Parameters: joint p1, p2,... , q1, q2 .... ,... n blocks  where pi,.., qi2;Example: joint 1, 0 , 1, 1If we need to multiply MFB(1,0) with MFB(1,1) we must follow the following steps:   1) Call MFB(1,0) and MFB 1, 1 .    2) For each output, make the following evalutions: g / g i ; fj/fj i   where i=1..n.        So from MFB(1,0) = f1 g1, 0, by using the previous evaluations, we have  f1g11, 0.          From MFB(1,1) = f1 g1, 1C f2 g1, 0 g0, 1, by using the previous evaluations we have             f1 g11, 1Cf2 g21, 0 g20, 1.   3) Compute the following cefficients:       r = p1Cq1C... !$ p2Cq2C... !$.....$ pnCqnC... !       s = p1!$p2!$...$q1!$q2!$... , in the example r = 2!$1! = 2 and s = 1!$0!$1!$1! = 1   4) Multiply the two previous terms as follows:       rsf1g11, 0 f1 g11, 1C f2 g21, 0 g20, 1 =        = 2 f1 g11, 0 f2 g21, 1C2 f1 g11, 0 f22 g21, 0 g20, 1The Codejoint d proclocal p1, p2, M, V;7 - 15> (4.2.3.2)(4.2.3.3)(4.2.3.1)> >        V  d `if` nargs = 1, args , args ;       p1 d mul add y, y = x !, x = Transpose V ;       p2 d mul x!, x = Flatten V ;      M  d max seq  add y, y = x , x = V ;     expand p1p2$mul   eval MFB op Vi , g = g i,                         seq fj = f ij, j = 1 ..M , i = 1 ..nargs ;  end:Examplesjoint 1, 0 , 1, 1 ;2 f1 g11, 0 f2 g21, 1C2 f1 g11, 0 f22 g21, 0 g20, 1joint 1, 1 , 2, 1 ;12 f1 g11, 1 f22 g21, 1 g21, 0C6 f1 g11, 1 f23 g21, 02  g20, 1C6 f1 g11, 1 f2 g22, 1C6 f1 g11, 1 f22 g22, 0 g20, 1C12 f12 g11, 0 g10, 1 f22 g21, 1 g21, 0C6 f12 g11, 0 g10, 1 f23 g21, 02  g20, 1C6 f12 g11, 0 g10, 1 f2 g22, 1C6 f12 g11, 0 g10, 1 f22 g22, 0 g20, 1joint 1, 0 , 1, 1 , 0, 1 ;4 f1 g11, 0 f3 g30, 1 f2 g21, 1C4 f1 g11, 0 f3 g30, 1 f22 g21, 0 g20, 1The function mkTIntroductionThis procedure gives a list of all subvectors having the same lenght of a vector V and such that their summation returns V. Parameters: mkT(V, n),  n2;Note: the output of V = v1,.., vr   is p1,.., pr ,..., q1,.., qr  such thatp1C..Cq1 = v1 . . . prC..Cqr = vrExample: Suppose to call mkt([1,1],2).1) First the function makeTab is called: makeTab(1,1)= a1 a2 , 1 , a1, a2 , 1 .2) From this list and for each block, we consider only the first subblock, that is a1 a2 , a1, a2 .3) Since the first block a1 a2  has cardinality one, we set a1 a2, null  as we have asked for two blocks (that is two subvectors, note that the second parameter is n=2).  So we have 8 - 15> > > > (4.3.3.2)(4.3.3.1)(4.3.3.3)1, 1 , 0, 0 . The second block a1, a2  has two elements and so we have [1,0],[0,1]4) The result is the list 1, 1 , 0, 0 , 1, 0 , 0, 1 .5) At the end, in order to have all lists, we need to permute the subblocks for each block, that is:1, 1 , 0, 01, 0 , 0, 10, 0 , 1, 10, 1 , 1, 0This function is used in the master function.The CodemkT d proc V, n    local vE, L, nV;    nVd nops V ;    vEd seq ai, i = 1 ..nV ;    Ld seq  `if` nops x1 % n, x1, NULL , x = makeTab  op V  ;    Ld seq seq seq degree y, z , z = vE , y = x , 0$nV $ nKnops x , x = L ;    Ld seq  op permute x , x = L ;  end:ExamplesmkT 1, 1 , 1 ;1, 1mkT 1, 1 , 2 ;1, 1 , 0, 0 , 0, 0 , 1, 1 , 1, 0 , 0, 1 , 0, 1 , 1, 0for L in mkT 2, 1 , 2  do print L ; od;1, 1 , 1, 01, 0 , 1, 12, 1 , 0, 00, 0 , 2, 12, 0 , 0, 10, 1 , 2, 0The Master Function UMFBIntroduction(Umbral Multivariate Fa`a di Bruno’s Formula)All previous steps are combined in this procedure. The UMFB function recalls the MFB function for the univariate and the multivariate case (parameter n=1) and allows us to construct the multivariate-multivariate case (parameter n>1).Parameters: UMFB(V, n), V= v1, v2 ... , n = #different g functions the output is the expansion of9 - 15> > > (4.4.3.1)(4.4.3.4)> (4.4.3.2)> (4.4.3.3) vh1C..Chrvx1h1 ...vxrhrf g 1 x1, .. , xr , .. , g n x1, .. , xrEsample: UMFB([1,3],2) = v3vy2 vx f g 1 x, y , g 2 x, yPseudocode:     UMFB(V, n)         if n = 1 then return( MFB(V) );         S := 0;         for v in mkT( V, n ) do               S := S + joint(v) {with f1if2j ..fnk/fi, j,.., k }        next for        return(S);      end:The CodeUMFB d proc V, n    local S, vE;    if n = 1 then return expand eval MFB op V , g = g1 ; fi;    vEd seq f i = 1, i = 1 ..n ;    Sd add joint  op x  , x = mkT V, n ;    add  f seq degree x, f i , i = 1 ..n $eval x, vE , x = S ;  end: ExamplesUnivariate with Univariate drdxr f g x  we have the same result of MFB function.UMFB 2 , 1 ;f2 g112Cf1 g12UMFB 3 , 1 ;f3 g113C3 f2 g11 g12C f1 g13Univariate with Multivariate vrvx1 ..vxr f g x1,.., xr  we have the same result of MFB function.UMFB 1, 1 , 1 ;f1 g11, 1Cf2 g11, 0 g10, 1UMFB 2, 1 , 1 ;2 f2 g11, 1 g11, 0Cf3 g11, 02  g10, 1Cf1 g12, 1Cf2 g12, 0 g10, 110 - 15(4.4.3.7)(4.4.3.5)> > > > (4.4.3.6)(4.4.3.8)Multivariate with Multivariate vrvx1 ..vxr f g1 x1,.., xr ,.., gn x1,.., xr .UMFB 1 , 2 ;f1, 0 g11Cf0, 1 g21UMFB 2 , 2 ;2 f1, 1 g11 g21C f2, 0 g112Cf1, 0 g12Cf0, 2 g212C f0, 1 g22UMFB 1 , 3 ;f1, 0, 0 g11Cf0, 1, 0 g21Cf0, 0, 1 g31UMFB 1, 1 , 2 ;f1, 0 g11, 1Cf2, 0 g11, 0 g10, 1Cf0, 1 g21, 1Cf0, 2 g21, 0 g20, 1Cf1, 1 g11, 0 g20, 1Cf1, 1 g10, 1 g21, 0Test ResultsIntroductionIn this section we analyze the accuracy of the results comparing the output obtained by using the Maple diff function (routine by which the multivariate Faà di Bruno’s formula is computable in Maple) .To this aim, we set: f#1,#2,...,#n = D1, 1, 1 ...2, 2, 2 ...n, n, n f g 1 x1,.., xr ,.., g n x1,.., xr    g k i,..j. = viC..C jvx1i .. vxrj g k x1,.., xr  In order to compare the results we have set up three functions: decoSub, deco and testUMFB.The decoSub function is an auxiliary routine useful for the deco function. The deco function takes the result of the function UMFB as input and gives an output equal to the Maple function diff. ThetestUMFB function checks the accuracy of the result.Parameter: testUMFB V, n, tDebug  where V, n are the same as the UMFB function and tDebug 2 {true, false} [set tDebug = true if you need to display also the output of the function diff]Example:  v2vx1vx2 f g x1, x2 , g x1, x2 = vvx1 g1 x1, x2  vvx2 g1 x1, x2  D1, 1 f g1 x1, x2 , g2 x1, x2 +vvx1 g1 x1, x2  vvx2 g2 x1, x2  D1, 2 f g1 x1, x2 , g2 x1, x2 +v2vx1 vx2 g1 x1, x2  D1 f g1 x1, x2 , g2 x1, x2 +vvx2 g1 x1, x2  vvx1 g2 x1, x2  D1, 2 f g1 x1, x2 , g2 x1, x2 +11 - 15> vvx1 g2 x1, x2  vvx2 g2 x1, x2  D2, 2 f g1 x1, x2 , g2 x1, x2 +v2vx1 vx2 g2 x1, x2  D2 f g1 x1, x2 , g2 x1, x2We get the same result from  UMFB 1, 1 , 2  = f1, 0 g11, 1C f2, 0 g11, 0 g10, 1C f0, 1 g21, 1Cf0, 2 g21, 0 g20, 1Cf1, 1 g11, 0 g20, 1Cf1, 1 g10, 1 g21, 0settingf2, 0 = D1, 1 f g1 x1, x2 , g2 x1, x2 ;f0, 2 = D2, 2 f g1 x1, x2 , g2 x1, x2 ;f1, 1 = D1, 2 f g1 x1, x2 , g2 x1, x2 ;f1, 0 = D1 f g1 x1, x2 , g2 x1, x2  ;f0, 1 = D2 f g1 x1, x2 , g2 x1, x2 ;        g11, 1 =v2vx1 vx2 g1 x1, x2 ;   g21, 1 =v2vx1 vx2 g2 x1, x2g11, 0 =vvx1 g1 x1, x2 ;    g21, 0 =vvx2 g2 x1, x2g10, 1 =vvx2 g1 x1, x2 ;    g20, 1 =vvx2 g2 x1, x2The codedecoSub d proc vE, vVars, nTot_x     local v, nTot_g, vG, vX, cE;     cEd coeffs vE ;     vd vEcE;      nTot_gd nops vVars K1;     vd seq  seq `if` has x, u , u, NULL , u = vVars , cE, op op x 1 ,op x 22, x= op v2 ;     vXd seq x j, j = 1 ..nTot_x ;     vGd seq g i  vX , i = 1 ..nTot_g ;     cE* mul `if` x1 = f, Dseq i$ x 3, i , i = 1 ..nTot_g f vG ,                 diff  x1 vX , seq x i$x3, i, i = 1 ..nops x3x4 , x = v ;end:12 - 15> (5.3.3)(5.3.1)> > (5.3.2)> > deco :=proclocal v, V, vG, nTot_g, nTot_x;V := `if` nargs = 1, indets args , indets op args 1 1 ;for v in V doif has v, 'f ' thennTot_g := nops op v ;breakend ifend do;for v in V doif not has v, 'f ' thennTot_x := nops op v ;breakend ifend do;vG := seq g || i, i = 1 ..nTot_g , f ;if nops V = 2 thendecoSub args, vG, nTot_xelseadd decoSub x, vG, nTot_x , x = op argsend ifend proc:testUMFB d proc vX, ng, tDebug     local R0, R1, R2, F, uX, st :     Fd f seq g i seq x j, j = 1 ..nops vX , i = 1 ..ng ;     uXd seq x j$vX j , j = 1 ..nops vX ;      R0dUMFB vX, ng :      R1d deco R0 :     R2d expand diff F, uX ;     if tDebug then print R0 ; print R2 ; fi;     if expand R1KR2 = 0 then print TEST_OK  else print ___ERROR___ ;fi; end:Execution SamplesUMFB 1, 1 , 1 ;f1 g11, 1Cf2 g11, 0 g10, 1Result1d deco % ;Result1 := D f g1 x1, x2  v2vx2 vx1  g1 x1, x2 CD2 f g1 x1,x2  vvx1  g1 x1, x2  vvx2  g1 x1, x2Result2d diff f g1 x1, x2 , x1, x2 ;Result2 := D f g1 x1, x2  v2vx2 vx1  g1 x1, x2 CD2 f g1 x1,13 - 15> (5.3.5)> (5.3.7)> > (5.3.4)(5.3.6)> x2  vvx1  g1 x1, x2  vvx2  g1 x1, x2evalb Result1 = Result2 ;truetestUMFB 1, 1 , 1, true ;f1 g11, 1Cf2 g11, 0 g10, 1D f g1 x1, x2  v2vx2 vx1  g1 x1, x2 CD2 f g1 x1, x2  vvx1  g1 x1,x2  vvx2  g1 x1, x2TEST_OKtestUMFB 1, 1 , 2, true ;f1, 0 g11, 1Cf2, 0 g11, 0 g10, 1Cf0, 1 g21, 1C f0, 2 g21, 0 g20, 1C f1, 1 g11, 0 g20, 1Cf1, 1 g10, 1 g21, 0D1 f g1 x1, x2 , g2 x1, x2  v2vx2 vx1  g1 x1, x2 CD1, 1 f g1 x1, x2 , g2 x1,x2  vvx1  g1 x1, x2  vvx2  g1 x1, x2 CD2 f g1 x1, x2 , g2 x1,x2  v2vx2 vx1  g2 x1, x2 CD2, 2 f g1 x1, x2 , g2 x1, x2  vvx1  g2 x1,x2  vvx2  g2 x1, x2 CD1, 2 f g1 x1, x2 , g2 x1, x2  vvx1  g1 x1,x2  vvx2  g2 x1, x2 CD1, 2 f g1 x1, x2 , g2 x1, x2  vvx2  g1 x1,x2  vvx1  g2 x1, x2TEST_OKtestUMFB 4, 3 , 2, false ;TEST_OKExecution TimeIn this section we analize the difference of the execution time between  the UMFB function and the Maple diff function.Notice: to perform an appropriate test, make a restart (using "!!!" for recall the entire worksheet)  to clear any cash areas. testTimedproc V, nlocal r1, r2, h, lx, st;h := f seq g || i seq x || j, j = 1 ..nops V , i = 1 ..n ;14 - 15(6.1)> lx := seq `$` x j, Vj , j = 1 ..nops V ;       std time ;r1dUMFB V, n ;print UMFB_Time = time  K st ;  print UMFB_Tot_Elements = nops op r1 ;       std time ;      r2d expand diff h, lx ;print DIFF_Time = time  K st ; st := time ;print UMFB_Tot_Elements = nops op r2 ; end proc:testTime 7, 6 , 2 ;UMFB_Time = 3.406UMFB_Tot_Elements = 60190DIFF_Time = 30.188UMFB_Tot_Elements = 60190ConclusionsBy using the classical umbral calculus, we provide a new Maple algorithm for computing the multivariate Faà di Bruno's formula for the following compositions: univariate with univariate, univariate with multivariate, multivariate with univariate, multivariate with the same multivariate, multivariate with different multivariates in any arbitrary number of components. The resulting algorithm is speeder of the routine MAPLE diff performing the same computations. References[1] Di Nardo E., G. Guarino, D. Senato (2011), A new algorithm for computing the multivariate Faà di Bruno’s formula, Appl. Math. Comp. doi:10.1016/j.amc.2011.01.001 [2] Di Nardo, E., Senato, D. (2006) An umbral setting for cumulants and factorial moments.European J. Combin. 27, No. 3, 394–413. (http://www.arxiv.org/abs/math/0412052)[3]  Di Nardo E., G. Guarino, D. Senato, Multiset Subdivision, source Maple algorithm located in www.maplesoft.com (http://www.maplesoft.com/applications/view.aspx?SID=33039)[4] Di Nardo E., G. Guarino, D. Senato (2008) An unifying framework for k-statistics, polykays and their generalizations. Bernoulli. Vol. 14(2), 440-468. (http://www.unibas.it/utenti/dinardo/BEJ6163290408.pdf)[5]  Di Nardo E., G. Guarino, D. Senato, A Maple algorithm for k-statistics, polykays and their multivariate generalization, source Maple algorithm located in www.maplesoft.com (http://www.maplesoft.com/applications/view.aspx?SID=33040)Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor theauthor are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. 15 - 15Contact the author for permission if you wish to use this application in for-profit activities

A new algorithm for computing the multivariate Faà di Bruno’s formula

https://iris.unito.it/bitstream/2318/1561389/2/MultFormMaple.pdf

A new algorithm for computing the multivariate Faà di Bruno’s formula

Abstract

Similar works

Full text

Available Versions

Institutional Research Information System University of Turin

A new algorithm for computing the multivariate Fa&#224; di Bruno&#8217;s formula

Abstract

Similar works

Full text

Available Versions

Institutional Research Information System University of Turin

A new algorithm for computing the multivariate Faà di Bruno’s formula