Difference Equations. Representation in "closed form" Formula solutions

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1 Difference-Equations-3-ClosedForm.nb Difference Equations - Part 3 - Representation in "closed form" Formula solutions Prof. Dr. J. Ziegenbalg Institut für Mathemati und Informati Pädagogische Hochschule Karlsruhe electronic mail: homepage: iegenbalg@ph-arlsruhe.de à References Dürr R. / J. Ziegenbalg: Mathemati für Computeranwendungen: Dnamische Proesse und ihre Mathematisierung durch Differenengleichungen, Schöningh Verlag, Paderborn 989 This boo will be quoted as [DZ 989] in the following text. Second edition: Mathemati für Computeranwendungen; Ferdinand Schöningh Verlag, Paderborn 989 Goldberg S.: Introduction to Difference Equations; John Wile, New Yor 98 Rommelfanger H.: Differenen- und Differentialgleichungen; B.I., Zürich 977 à The annuit equation In the introduction, the annuit equation was defined as the difference equation + = A ÿ + B ü Implementation in Mathematica Annuit@0_, A_, B_, _D := Module@8 = 0, i = 0<, <, i = i + ; = A + BD; Return@D D

2 Difference-Equations-3-ClosedForm.nb 0000, 4D , id<, 8i,, <D 88, 9000.<, 8, 8970.<, 83, 8437.<, 84, <, 8, 737.8<, 86, <, 87, 990.<, 88, 4.<, 89, <, 80, 370.<, 8, 8966.<, 8, 044.4<, 83, 43.<, 84, <, 8, 789.8<< % êê TableForm Remove@0,, A, BD; Table@8i, Annuit@0, A, B, id<, 8i,, 0<D êê TableForm B+ A0 B+ A HB + A0L 3 B+ A HB + A HB + A0LL 4 B+ A HB + A HB + A HB + A0LLL B+ A HB + A HB + A HB + A HB + A0LLLL 6 B+ A HB + A HB + A HB + A HB + A HB + A0LLLLL 7 B+ A HB + A HB + A HB + A HB + A HB + A HB + A0LLLLLL 8 B+ A HB + A HB + A HB + A HB + A HB + A HB + A HB + A0LLLLLLL 9 B+ A HB + A HB + A HB + A HB + A HB + A HB + A HB + A HB + A0LLLLLLLL 0 B + A HB + A HB + A HB + A HB + A HB + A HB + A HB + A HB + A HB + A0LLLLLLLLL Table@8i, Annuit@0, A, B, id<, 8i,, 0<D êê Simplif êê TableForm B+ A0 B+ A HB + A0L 3 B+ A HB + A HB + A0LL 4 H + A + A + A 3 L B + A 4 0 H + A + A + A 3 + A 4 L B + A 0 6 H + A + A + A 3 + A 4 + A L B + A H + A + A + A 3 + A 4 + A + A 6 L B + A H + A + A + A 3 + A 4 + A + A 6 + A 7 L B + A H + A + A + A 3 + A 4 + A + A 6 + A 7 + A 8 L B + A H + A + A + A 3 + A 4 + A + A 6 + A 7 + A 8 + A 9 L B + A 0 0 H + A + A + A 3 + A 4 + A + A 6 + A 7 + A 8 + A 9 L H AL êê Simplif A 0

3 Difference-Equations-3-ClosedForm.nb 3 A i i=0 + A + + A i j A i H AL i=0 A + êê Simplif ü A more "difference equation" lie stle of A_, B_, _D := Module@8 = 0, i = 0<, <, i = i + ; = A + BD; Return@D 0000, 4D , id<, 8i,, <D 88, 9000.<, 8, 8970.<, 83, 8437.<, 84, <, 8, 737.8<, 86, <, 87, 990.<, 88, 4.<, 89, <, 80, 370.<, 8, 8966.<, 8, 044.4<, 83, 43.<, 84, <, 8, 789.8<< % êê TableForm

4 Difference-Equations-3-ClosedForm.nb 4 ü Representations / solutions in "closed form" Theorem (Annuit equation in closed form - AECF) (cf. [DZ 989], Sat 7., page 48) The term of the annuit equation + = A ÿ + B can be expressed in the following wa (a) = H + A + A + A 3 + A A - L ÿ B + A ÿ 0 If A then (a) = A - ÅÅÅÅÅÅÅÅÅÅÅÅÅ A- ÿ B + A ÿ 0 If A = then (a) = ÿ B + A ÿ 0 Exercise: Prove the above theorem b formall appling (mathematical) induction. Corollar (a) The term of the geometric sequence + = A ÿ can be expressed in the following wa = A ÿ 0 (a) The term of the arithmetic sequence + = + B can be expressed in the following wa = ÿ B + 0 ü Some (philosophical) remars on methodolog and terminolog In the literature on difference equations the above theorem (in particular variant a) is often expressed in a wording according to which the closed form representation = A - ÅÅÅÅÅÅÅÅÅÅÅÅÅ A- ÿ B + A ÿ 0 is the "solution" of the difference equation + = A ÿ + B. It is, however, debatable, in which sense this is the case and for what purposes the closed form representation is more adequate than the original (recursive) version (cf. J. Ziegenbalg: "Formula versus Algorithm"; paper presented at the the conference The Origins of Algebra: From al-khwarimi to Descartes, Universitat Pompeu Fabra (UPF) and Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, March 7-9, 003) Aspects subject to discussion are:. Computational efficienc. Cognitive efficienc 3. Historical aspects

5 Difference-Equations-3-ClosedForm.nb The representations (a) and (a) in theorem AECF are often called representations in closed form or solutions in closed form. The terminolog "closed form" suggests that these representations are non-recursive. But at a closer loo this turns out to be onl part of the truth, because in the evaluation, for instance, of a term lie A recursion comes into the game, again. In spite of all this, in the following text the standard terminolog is adopted and in particular the term = A - ÅÅÅÅÅÅÅÅÅÅÅÅÅ A- ÿ B + A ÿ 0 will be called a "solution" of the difference equation + = A ÿ + B. Similarl, the term "solution" will also be used in this wa for other (more general) difference equations. ü Woring with the Mathematica Pacage "Discrete Math" The "add-on" pacage "Discrete Math" distributed with Mathematica contains the function RSolve (for "recurrence solve") b which solutions in closed form can convenientl be obtained for some tpes of recursive equations. The next line shows how to load the RSolve function. << DiscreteMath`RSolve`? RSolve RSolve@eqn, a, nd solves a recurrence equation for the function a, with independent variable n. RSolve@8eqn, eqn,...<, 8a, a,...<, nd solves a list of recurrence equations. Mehr The next line shows an example of how to use the RSolve function. RSolve@@ + D + D 99@D H + A L B + H + AL A C@D == + A In the last expression C[] is a constant which can be determined b adjusting the general form to initial values. The next line shows an evaluation, resulting in [0]. ReplaceAllA H + A L B + H + AL A C@D + A C@D, 0E The same call in a different sntax: H + A L B + H + AL A C@D + A C@D ê. 0 Above, we used the definition [0] = 0. So the constant C[] is just our initial value 0. In the next example, the initial conditions are specified within RSolve.

6 Difference-Equations-3-ClosedForm.nb 6 RSolve@8@ + D + == D 99@D H + A L B + H + AL A 0 == + A à The generalied Fibonacci equation ü A remar on the historical development In his Liber abaci (0), Leonardo of Pisa (called Fibonacci, ca. 70-0) formulated a problem giving rise to the following famous sequence of numbers now called the "Fibonacci" numbers:,,, 3,, 8, 3,, 34,, 89,... Its most important propert is that ever member of the sequence is the sum of its two immediate predecessors (except for the initial values): F + = F + + F It too several centuries until J. P. M. Binet (786-86) finall presented the following formula for the Fibonacci numbers: F = ÅÅÅÅÅÅÅÅÅÅ J + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N - ÅÅÅÅÅÅÅÅÅÅ J - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N The formula nowadas is nown as Binet's formula. We will now develop this formula within the framewor of difference equations. In case that the reader is doubtful that the formula is correct (which is perfectl plausible, regarding the complexit of the formula and in particular the embedded root expressions), here is a preliminar test: TableANA i j + i j E, 8,, <E 8.,.,., 3.,., 8., 3.,., 34.,., 89., 44., 33., 377., 60., 987., 97., 84., 48., 676., 0946., 77., 867., , 70.< ü Terminolog In this section we will consider the following linear difference equation of oder with constant coefficients and constant inhomogeneit A ÿ + + A ÿ + + A 0 ÿ = 0 (where A 0, A and A are fixed constants; A 0) Since A 0 we can divide this equation b A and obtain the somewhat simpler but equivalent form + + a ÿ + + a 0 ÿ = 0 (* GFE *) This equation clearl generalies the Fibonacci equation; we will, therefore, call it the generalied Fibonacci equation (GFE) for short.

7 Difference-Equations-3-ClosedForm.nb 7 ü Appling standard methodolog Usuall in mathematics (and elsewhere) it is more difficult to solve a generalied form of a specific problem than to solve the specific problem itself, for the solution of the generalied problem contains the solution of the specific problem. Sometimes, however, the solution of the more general problem turns out to be easier than the solution of the specific problem thus supporting the dictum "be wise, generalie". Obtaining a closed form representation for the Fibonacci numbers is a striing illustration of this fact. It will turn out to be simpler and more natural to solve the generalied Fibonacci equation than the original (special) Fibonacci equation. A general strateg in mathematical problem solving is to tr to reduce a new, unnown situation to well-nown cases. In this wa, we will tr to convert the generalied Fibonacci equation (of order ) into two annuit equations (of order ). If we succeed, we can hope combine the closed form representations of these annuit equations into a closed form representation of the generalied Fibonacci equation. ü First attempt The generalied Fibonacci equation might, for instance, be thought of as the "addition" of the following two first-order equations: + + ÅÅÅÅ a ÿ + = 0 ÅÅÅÅÅ a ÿ + + a 0 ÿ = 0 Exercise: Convert each of these two first-order equations into the standard form for geometric sequences and show that the, in general, have no common solution (i.e. no common closed form representation). ü Second attempt: Introducing new parameters for greater flexibilit (in this case: introducing a "tuning" parameter) Due to this result we have to handle the decomposition in a slightl more subtle wa b introducing an extra parameter called t (for "tuning") in the following wa. + + Ha + tl ÿ + = 0 -t ÿ + + a 0 ÿ = 0 Still, GFE can be thought of as being the sum of these first-order equations. If, b choosing a suitable "tuning" value for t, we can mae these two first-order equations identical, then the will have the same closed form representations and we can tr to combine their individual solutions into a solution for the generalied Fibonacci equation. Written in the "standard" form for geometric sequences the last two first-order equations read

8 Difference-Equations-3-ClosedForm.nb 8 ü Tuning process - with the goal to mae the two equations identical + =-Ha + tl ÿ + (* GS- *) + = ÅÅÅÅÅÅ a 0 ÿ t (* GS- *) These difference equations for geometric sequences are identical if their coefficients -Ha + tl and ÅÅÅÅÅÅ a 0 t (The "index-shift" b is irrelevant, since the equations are valid for all values of ). A necessar condition for equalit, hence, is are equal. -Ha + tl = ÅÅÅÅÅÅ a 0 t i.e. t + a ÿ t + a 0 = 0 ü The characteristic polnomial Thus, the above geometric sequences are identical if the "tuning" parameter t satisfies the so-called characteristic equation of (* GFE *): x + a ÿ x + a 0 = 0 We finall obtain the tuning parameters t = -a +!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 t = -a -!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 ü Solutions - b appling the results on geometric series Thus, the geometric sequences adding up to (* GFE *) are. B using the root t : + =-Ja + -a +!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N ÿ + a 0 + = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ -a +!!!!!!!!!!!!!!! ÿ a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 0. B using the root t : + =-Ja + -a -!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N ÿ + a 0 + = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ -a -!!!!!!!!!!!!!!! ÿ a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 0 Exercise: Simplif these formulae.

9 Difference-Equations-3-ClosedForm.nb 9 ü Vieta's formulae Excursion: Vieta's formulae The tuning parameters t and t, being the roots of the quadratic equation x + a ÿ x + a 0 = 0 must satisf Vieta's equations: t + t =-a and t ÿ t = a 0 Thus, b substituting either t or t for t and appling Vieta's formulae we can rewrite (* GS- *) and (GS- *) in the following wa: + = t ÿ + (* GS-. *) + = t ÿ + (* GS-. *) + = t ÿ (* GS-. *) + = t ÿ (* GS-. *) B pure combinatorics, substituting two roots into two equations formall gives four cases, but b algebra (Vieta) these melt down to two essentiall different cases. ü Results Theorem (solutions of the generalied Fibonacci equation) (cf. [DZ 989], Sat 3., page 90) The generalied Fibonacci equation + + a ÿ + + a 0 ÿ = 0 has the following "solutions" (i.e. closed form representations): (a) and (a) = J -a +!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N = J -a -!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N Obtaining more solutions: Theorem (combining solutions) If the sequences Hu L =0,..., and Hv L =0,..., are solutions of the generalied Fibonacci equation

10 Difference-Equations-3-ClosedForm.nb a ÿ + + a 0 ÿ = 0 then their "sum" (a) Hu L =0,..., Hv L =0,..., := Hu + v L =0,..., and for an real number C the "scalar multiple" (a) C Ÿ Hu L =0,..., := HC ÿ u L =0,..., also are solutions of the generalied Fibonacci equation. Proof: Exercise Henceforth we will use the simpler operation smbols + and instead of and Ÿ. Corollar (linear combination of solutions): If the sequences Hu L =0,..., and Hv L =0,..., are solutions of the generalied Fibonacci equation + + a ÿ + + a 0 ÿ = 0 then for an real numbers C and C the "linear combination" C ÿ Hu L =0,..., + C ÿ Hv L =0,..., also are solutions of the generalied Fibonacci equation. Corollar The generalied Fibonacci equation + + a ÿ + + a 0 ÿ = 0 has the following "solutions" (i.e. closed form representations): C ÿ J -a +!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N + C ÿ J -a -!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N where C and C are arbitrar real (or complex) numbers. (* S-GFE *) In other words: The set of all solutions of the generalied Fibonacci equation is a vector space over a suitable scalar field (usuall the field the coefficients are taen from); cf. [DZ 989], Sat 4., page 9. Furthermore, it is not difficult to see that the dimension of this vector space is and that, in case the solutions t and t of GFE's characteristic equation do not coincide, then the sequences Ht L and Ht L are a basis of this vector space. ü Initial values The above results were valid independent of an initial values 0 and of the GFE. Let us now assume that the solution (* S-GFE *), additionall, is to satisf these initial values. Then for = 0 and = the following two linear equations will have to be satisfied b C and C : Corollar The initial values of the generalied Fibonacci equation + + a ÿ + + a 0 ÿ = 0 can be expressed in the following wa:

11 Difference-Equations-3-ClosedForm.nb C ÿ J -a +!!!!!!!!!!!!!! a 0-4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N + C ÿ J -a -!!!!!!!!!!!!!! a 0-4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N = 0 (* LE- *) i.e. C ÿ J -a +!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N + C ÿ J -a -!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N = (* LE- *) C + C = 0 C ÿ -a +!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 + C ÿ -a -!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 = Notwithstanding the algebraic complexit of these equations, the are two simple linear equations in the two unnowns C and C which can be solved b straightforward algebraic procedures. Exercise: Show that and C = a 0 "################## 4 a 0 +a 0 "################## 4 a 0 +a C = a 0 "################## 4 a 0 +a 0 + "################## 4a 0 +a are solutions of (* LE- *) and (* LE- *). ü Appling the results to the sequence of the standard Fibonacci numbers - Binet's formula The sequence of the standard Fibonacci numbers, equivalentl either starting with index 0 or index is given b Specialiing from GFE, its parameters are: a =- and a 0 =-. Hence, the homogeneous equation = 0 has the "general" solution i.e. = C ÿ J -a +!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N + C ÿ J -a -!!!!!!!!!!!!!! a -4 a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ 0 N = C ÿ J + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N + C ÿ J - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N

12 Difference-Equations-3-ClosedForm.nb ü Putting it all together using Mathematica's "smbol manipulation" features i a + "################ a ###### 4 a i 0 Solve A9C + C == 0, C j + C a "################ a ###### 4 a 0 j == =, 8C,C <E 99C a 0 "################ 4 a ###### 0 + a 0 "################ 4 a ######,C a 0 "################ 4a ###### 0 + a a "################ 4 a ###### == 0 + a C = a 0 "################ 4 a ###### 0 + a 0 "################ 4 a ###### 0 + a a 0 "################ 4a ###### 0 + a 0 "################ 4a ###### 0 + a C = a 0 "################ 4 a ###### 0 + a 0 "################ 4 a ###### 0 + a ê. 8a 0, a, 0 0, < C = a 0 "################ 4 a ###### 0 + a 0 + "################ 4 a ###### 0 + a a 0 "################ 4 a ###### 0 + a 0 + "################ 4a ###### 0 + a C = a 0 "################ 4 a ###### 0 + a 0 + "################ 4 a ###### 0 + a ê. 8a 0, a, 0 0, < ü Binet's Formula Theorem (Binet) The Fibonacci equation + = + + has the following "solution" (i.e. closed form representations): = ÅÅÅÅÅÅÅÅÅÅ ÿ J + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N - ÅÅÅÅÅÅÅÅÅÅ ÿ J - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ N Chec

13 Difference-Equations-3-ClosedForm.nb 3 = i j + i j I I MM + I I + MM TableA9, i j + i j 980, 0<, 9, + + =, =, 8, 0, <E I M I + M I M 3 I + M 3 9, =, 93, =, I M 4 I + M 4 I M I + M 94, =, 9, == TableA9, i j + i j êê N=, 8, 0, <E 880, 0.<, 8,.<, 8,.<, 83,.<, 84, 3.<, 8,.<, 86, 8.<, 87, 3.<, 88,.<, 89, 34.<, 80,.<, 8, 89.<, 8, 44.<, 83, 33.<, 84, 377.<, 8, 60.<, 86, 987.<, 87, 97.<, 88, 84.<, 89, 48.<, 80, 676.<, 8, 0946.<, 8, 77.<, 83, 867.<, 84, <, 8, 70.<< % êê TableForm

14 Difference-Equations-3-ClosedForm.nb 4 ü Appling genuine computeralgebra features TableA9, i j + i j =, 8, 0, 0<E 980, 0<, 9, + + I M I + M =, 9, =, I M 3 I + M 3 I M 4 I + M 4 93, =, 94, =, I M I + M I M 6 I + M 6 9, =, 96, =, I M 7 I + M 7 I M 8 I + M 8 97, =, 98, =, I M 9 I + M 9 I M 0 I + M 0 99, + =, 90, == % êê Simplif 880, 0<, 8, <, 8, <, 83, <, 84, 3<, 8, <, 86, 8<, 87, 3<, 88, <, 89, 34<, 80, << ü RSolve for GFE RSolving the GFE: Remove@, 0,, a0, ad RSolve@8@ + D + + D + == == D 99@D!!!!!!!!!!!!! 4 a0+!!!!!!!!! a i j i j J a + "################ 4 a0+ ######### a N + + J a "################ ######### 4 a0+ a N Ja + "################ 4 a0+ ######### a N 0 + i j J a "################ 4a0+ ######### a N + J a + "################ 4 a0+ ######### a N Ha 0 + L == RSolving the original Fibonacci equation: RSolve@8@ + + == == D 99@D I I MM I I + MM == à Auxiliar stuff