An Algorithm to Calculate the Inverse Matrix if an Infinite Matrix

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The The 12th 12th International International Conference Conference Interdisciplinarity Interdisciplinarity in in Engineering Engineering

An Algorithm to Calculate the Inverse Matrix if an Infinite Matrix B´eela la Finta FintaConference 2017, MESIC 2017, 28-30 June Manufacturing Engineering Society International B´ 2017, Vigo (Pontevedra), Spain “Petru Maior” Maior” University University of of Tˆ Tˆa argu rgu Mures Mures “Petru N. Iorga Iorga street street nr. nr. 1, 1, 540088, 540088, Romania Romania N.

Costing models for capacity optimization in Industry 4.0: Trade-off between used capacity and operational efficiency Abstract Abstract

a a,* left inverse matrix b b The an calculate an A.give Santana , P.to A. inverse Zaninmatrix , R.of The purpose purpose of of this this paper paper is is to to give an algorithm algorithm toAfonso calculate the the, left ofWernke an infinite infinite matrix matrix using using the the extension extension to to infinite matrices of the LU matrix factorization. We give an example for the numerical solution of the heat equation. infinite matrices of the LU matrix factorization. We give an example for the numerical solution of the heat equation. a

University of Minho, 4800-058 Guimarães, Portugal

Unochapecó, 89809-000 Chapecó, SC, Brazil © The Authors. Published by Elsevier bLtd. cc 2019  2018  2018 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. This is is an an open open access access article article under under the the CC CC BY-NC-ND BY-NC-ND license license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and and peer-review peer-review under under responsibility responsibility of of the the 12th 12th International International ConferenceInterdisciplinarity Interdisciplinarityin inEngineering. Engineering. Selection Selection and peer-review under responsibility of the 12th International Conference Conference Interdisciplinarity in Engineering.

Abstract Keywords: Infinite matrix, matrix, operations operations with with infinite infinite matrices, matrices, LU LU matrix matrix factorization, factorization, infinite infinite lower lower triangular triangular matrix, matrix, infinite infinite upper upper triangular triangular Keywords: Infinite matrix. matrix.

Under the concept of "Industry 4.0", production processes will be pushed to be increasingly interconnected, information based on a real time basis and, necessarily, much more efficient. In this context, capacity optimization goes beyond the traditional aim of capacity maximization, contributing also for organization’s profitability and value. 1. Introduction Introduction Indeed, lean management and continuous improvement approaches suggest capacity optimization instead of 1. maximization. The study of capacity optimization and costing models is an important research topic that deserves Let aa wire diameter mm 30 is In we consider contributions from both thewith practical and 33theoretical perspectives. This paper presents and case discusses a mathematical Let us us consider consider wire with diameter mm and and length length 30 m, m, which which is heated. heated. In this this case we can can consider the the following mathematical model of ofbased the heat heat equation:costing −u”(x) models + c(x) c(x) ·· (ABC u(x) = =and (x), where xx A ∈ generic [0, +∞), +∞),model and u(0) u(0) =been µ00 .. model for mathematical capacity management on different TDABC). has= following model the equation: −u”(x) + u(x) ff (x), where ∈ [0, and µ ∈and R is istoaadesign given strategies positive real real value. Inmaximization order to to obtain obtain the numerical numerical Here c, c, ff ::and [0,it+∞) +∞) are given functions and µ00 ∈ developed wasare usedgiven to analyze idle and capacity towards theIn of organization’s R given positive value. order the Here [0, functions µ solution of this this problem we must must consider the thevsdiscrete discrete form of of the heat heat is equation. For this this reason we consider consider fixed value. The trade-off capacity maximization operational efficiency highlighted andreason it is shown that capacity solution of problem we consider form the equation. For we aa fixed small positive positivemight real value value h∈ ∈ R, R, h h> > inefficiency. 0 and and the the knots knots xxii = = ii ·· h, h, ii ∈ ∈ N. N. We We denote denote uuii = = u(x u(xii ), ), ccii = = c(x c(xii ), ), ffii = = ff (x (xii )) for for optimization hide operational small real h 0 =B.V. u(x00 )) = = u(0) u(0) = =µ µ00 and and for for every every ii ∈ ∈N N− − {0} {0} we we obtain obtain the the discrete discrete equation equation every ∈ N. N.Authors. In this this Published way we we obtain obtain u00 = © 2017ii The by Elsevier u(x every ∈ In way u −2·ui +u +ui−1 22 + c ) · u =Engineering i+1 −2·u i−1 Peer-review under theii scientific the(( Manufacturing International + cciiresponsibility uii = = ffii .. So Sooffor for =1 1 we we committee get − − h1122 ·· u uof22 + + f + 11 · µSociety and for i ∈ N, i ≥Conference 2 we have − uui+1 2i + ·· u = get − hh22 + c11 ) · u11 = f11 + hh22 · µ00 and for i ∈ N, i ≥ 2 we have h hh2 11 2 1 2017. 22 + ci ) · ui − 12 · ui−1 = fi . These equations we can write in matrix form A · x = b, where A = (ai j )i, j∈N−{0} is − · u + ( i+1 2 − hh2 · ui+1 + ( hh2 + ci ) · ui − hh2 · ui−1 = fi . These equations we can write in matrix form A · x = b, where A = (ai j )i, j∈N−{0} is 2 1 an infinite matrix with infinite, but numerable rows columns such 11 = 12 = an infiniteCost matrix with infinite, butCapacity numerable rows and and columns such that that aaEfficiency = hh222 + + cc11 ,, aa12 =− − hh122 and and for for ii ∈ ∈ N, N, ii ≥ ≥ 22 11 Keywords: Models; ABC; TDABC; Management; Idle Capacity; Operational 1 2 1 2 ai,i−1 = ai,i+1 = =− − h22 and and a ai,i = + c . The other elements of the infinite matrix A are equal 0, so we obtain an infinite a i,i−1 = ai,i+1 i,i = hh22 + cii . The other elements of the infinite matrix A are equal 0, so we obtain an infinite h = uuii ,, where where ii ∈ ∈N N \\ {0} {0} and and the the matrix matrix tridiagonal matrix matrix A. A. The The matrix matrix xx is is an an infinite infinite column column matrix matrix with with elements elements xxii = tridiagonal ∈N N− − {0}, {0}, where where bb11 = = ff11 + + h1122 ·· µ µ00 and and bbii = = ffii for for every every ii ∈ ∈N N and and b 1. is Introduction also an an infinite infinite column column matrix matrix with with elements elements b bii ,, ii ∈ b is also h ≥ 2. 2. We We mention mention that that we we multiply multiply the the rows rows of of the the matrix matrix A A with with the the column column matrix matrix xx and and we we obtain obtain numerical numerical series series ii ≥ The cost of idle capacity is a fundamental information for companies and their management of extreme importance in∗modern production systems. In general, it is defined as unused capacity or production potential and can be measured ∗ Corresponding author. Tel.: +40-265-262275; fax: Corresponding fax: +40-265-262275. +40-265-262275. in several ways:author. tonsTel.: of +40-265-262275; production, available hours of manufacturing, etc. The management of the idle capacity E-mail address: [email protected] E-mail address: [email protected] * Paulo Afonso. Tel.: +351 253 510 761; fax: +351 253 604 741 E-mail address: [email protected]

2351-9789 ©c 2017 The Authors. Published by Elsevier B.V. 2351-9789 c 2018 2351-9789   2018 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. Peer-review under responsibility of the scientific of the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2019 The Authors. Published bycommittee Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the 12th International Conference Interdisciplinarity in Selection under responsibility of the Conference Interdisciplinarity in Engineering. Engineering. Selectionand andpeer-review peer-review under responsibility of12th the International 12th International Conference Interdisciplinarity in Engineering. 10.1016/j.promfg.2019.02.265

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Béla Finta / Procedia Manufacturing 32 (2019) 643–646 B´ela Finta / Procedia Manufacturing 00 (2018) 000–000

whose sums are equal with the corresponding elements of the column matrix b. If we can calculate the left inverse matrix A−1 of the matrix A, then we multiply from left the matrix equation A · x = b with the matrix A−1 and we obtain x = A−1 · b, exactly the numerical solution of the heat equation. 2. Main part First of all we denote the set N − {0} with N∗ . Next we give some definitions. A is called a real (complex) infinite matrix, if has infinite, but numerable rows and columns, i.e. A = (ai j )i, j∈N∗ , where the elements ai j ∈ R are real numbers (ai j ∈ C are complex numbers) for every i, j ∈ N∗ . The infinite matrices A and B = (bi j )i, j∈N∗ are equal, i.e. A = B, if ai j = bi j for every i, j ∈ N∗ . If A and B are two real (complex) infinite matrices, then we can define the sum of these matrices: A + B = (ai j + bi j )i, j∈N∗ and multiplication by real (complex) scalars: α · A = (α · ai j )i, j∈N∗ , where  α ∈ R (α ∈ C). The product of these matrices we define like A · B = ( ∞ k=1 aik · bk j )i, j∈N∗ , where the product matrix ∞ exists if and only if all the series k=1 aik · bk j are convergent series for every i, j ∈ N∗ . In other case the product matrix does not exist. It is known the LU factorization for finite matrices, see for example [1]. First of all we extend the LU factorization from finite matrices to infinite matrices. The infinite matrix L = (li j )i, j∈N∗ we call infinite lower triangular matrix, if li j = 0 for every i, j ∈ N∗ and i < j. The infinite matrix U = (ui j )i, j∈N∗ we call infinite upper triangular matrix, if ui j = 0 for every i, j ∈ N∗ and i > j. The next result we showed in [2]. Proposition 1. For the infinite matrix A there exists the LU factorization, i.e. there exist L infinite lower triangular matrix and U infinite upper triangular matrix, such that A = L · U. If we choose the elements of the principal diagonal of L equals one, i.e. lkk = 1 for every k ∈ N∗ , then the LU infinite matrix factorization is unique determined. Proof. We use the mathematical induction method. First we determine the first column of the infinite matrix L and the first row of the infinite matrix U using the following relations obtained by matrix multiplication of the rows of L with the columns of U: 1 · u11 = a11 and for every i ∈ N, i ≥ 2 we have li1 · u11 = ai1 . At the same time for every j ∈ N, j ≥ 2 we get 1 · u1 j = a1 j . We suppose that a11  0, so u11 = a11 with u11  0, for i ≥ 2 li1 = ua11i1 and for j ≥ 2 u1 j = a1 j . Next we suppose that we calculated the elements of L from the first n − 1 columns and the elements of U from the first n − 1 rows with n ≥ 2. By the mathematical induction step next we determine the elements of L from the column n: lin for i ≥ n + 1 and the elements of U from the row n: un j for j ≥ n using the following relations obtained with   · unn = ann , so unn = ann − n−1 matrix multiplication: n−1 k=1 lnk · ukn + 1  k=1 lnk · ukn . We suppose that unn  0. For i ≥ n + 1 n   ain − n−1 k=1 lik ·ukn we have k=1 lik · ukn = ain , so lin = . For j ≥ n + 1 we have nk=1 lnk · uk j = an j , so un j = an j − n−1 k=1 lnk · uk j . unn The infinite matrix E is an identity matrix, if for every infinite matrix A we have E · A = A · E = A. It is immediately that the infinite matrix I∞ is an identity matrix, if the principal diagonal elements are equal one and the other elements are equal zero. We can show that the infinite identity matrix is unique. Indeed, from E · A = A · E = A for A = I∞ results E · I∞ = I∞ · E = I∞ , so E = I∞ . Next we want to obtain the inverse matrix of the infinite matrix A. Let A = L · U the LU factorization for the infinite matrix A. We will determine the elements of the lower triangular infinite matrix L’ such that L · L = I∞ and the elements of the upper triangular infinite matrix U’ such that U  · U = I∞ . We suppose that the product of infinite matrices U  · L · L · U is associative, so U  · L · L · U = I∞ . This means that the infinite matrix A = L · U admits the left inverse matrix A−1 = U  · L . Here we suppose that there exists the product of the infinite matrices U’ and L’. Proposition 2. There exists a unique lower triangular infinite matrix L = (li j )i, j∈N∗ such that L · L = I∞ .   Proof. We use the mathematical induction method. We have l11 · l11 = 1 so l11 = 1. We multiply the first row of L’ with the columns of L and we obtain the elements 1, 0, 0, . . . , 0, . . . which is exactly the first row of the matrix I∞ . We     ·1+l22 ·l21 = 0, l21 ·0+l22 ·l22 = 1, multiply the second row of L’ with the columns of L and we receive the equalities l21      l21 · 0 + l22 · 0 = 0, and so an. In this way we get the second row of I∞ . From l21 · 0 + l22 · l22 = 1 results l22 = 1 and    · 1 + l22 · l21 = 0, we get l21 = −l21 . from l21



Béla Finta / Procedia Manufacturing 32 (2019) 643–646 B´ela Finta / Procedia Manufacturing 00 (2018) 000–000

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We suppose that we obtained the first n − 1 rows of L’ with n ≥ 3 and next we determine the row n of L’, by matrix multiplication of the row n of L’ with the columns of L:

     · 1 + ln2 · l21 + ln3 · l31 + · · · + ln,n−1 · ln−1,1 + lnn · ln1 = 0 ln1  ln1  ln1

·0+

·0+

 ln2  ln2

·1+

·0+

 ln3  ln3

· l32 + · · · +

 ln,n−1

· 1 + ··· +

 ln,n−1

· 0 + ··· +

 ln,n−1  ln,n−1

· ln−1,2 +

· ln−1,3 +

 lnn

 lnn

· ln2 = 0

· ln3 = 0

...      · 1 + lnn · ln,n−1 = 0 · 0 + ln3 · 0 + · · · + ln,n−1 · 0 + ln2 ln1

 ln1  ln1

...

·0+ ·0+

 ln2  ln2

·0+ ·0+

 ln3  ln3

· 0 + ··· +

·0+ ·0+

 lnn  lnn

·1=1 ·0=0

(1) (2) (3) (4) (5) (6)

  In this way we get the row n of I∞ . From the equation 5 results lnn = 1, from the equation 4 we get ln,n−1 , and so on,    , from the equation 2 ln2 , and finally from the equation 1 ln1 , respectively. and we obtain from the equation 3 ln3

Proposition 3. There exists a unique upper triangular infinite matrix U  = (ui j )i, j∈N∗ such that U  · U = I∞ . Proof. We use the mathematical induction method. We multiply the first row of U’ with the columns of U and we get the first row of I∞ : from u11 ·u11 = 1 we find u11 , from u11 ·u12 +u12 ·u22 = 0 we find u12 . We suppose that we obtained u11 , u12 , . . . , u1,n−1 and we determine u1n from the equality u11 · u1n + u12 · u2n + · · · + u1,n−1 · un−1,n + u1n · unn = 0. These values exist because ukk  0 for every k ∈ N∗ . Next we suppose that we obtained for the matrix U’ the first n − 1 rows, n ≥ 2 and next we find the elements of the row n from the matrix U’. We multiply the row n of U’ with the first, second, and n − 1 columns of U and we get in each case 0. We multiply the row n of U’ with the column n of U: unn · unn = 1 and we find unn . We suppose that ukk  0 for k ∈ N∗ . We multiply the row n of U’ with the column n + 1 of U: unn · un,n+1 + un,n+1 · un+1,n+1 = 0 and we obtain un,n+1 . We multiply the row n of U’ with the column n + 2 of U: unn · un,n+2 + un,n+1 · un+1,n+2 + un,n+2 · un+2,n+2 = 0 and we find un,n+2 , and so on. In this way we determine step by step the elements of the row n of U’. 3. Applications At the end we show a concrete example. Let A be an infinite matrix with elements given in the following way: a11 = 2, for k ≥ 2 akk = 1, for k ≥ 1 ak,k+1 = 1, for k ≥ 2 ak,k−1 = −2, and the other elements are equal zero. Using proposition 1 we obtain the infinite matrices L and U with elements: for k ≥ 1 lkk = 1, for k ≥ 2 lk,k−1 = −1, and the other elements are zero, for k ≥ 1 ukk = 2, for k ≥ 1 uk,k+1 = 1, and the other elements are zero. Using proposition 2 we calculate the infinite lower triangular matrix L’ with elements: for i ≥ j ≥ 1 li j = 1 and the other elements are i+ j

, equal zero. Using proposition 3 we get the infinite upper triangular matrix U’ with elements: for 1 ≤ i ≤ j ui j = (−1) 2 j−i+1 and the other elements are equal zero. We obtain the infinite inverse matrix A−1 = U  · L with elements: for i > j ≥ 1 i+ j . We can verify easily that A−1 · A = I∞ , so A−1 is the left inverse of the infinite ai j = 13 and for 1 ≤ i ≤ j ai j = 13 · (−1) 2 j−i −1 matrix A. We can observe also A · A = I∞ , so A−1 is the right inverse of the infinite matrix A, too. We mention that in the above presented algorithm is true also for finite matrices and we can obtain the inverse matrix for a given invertible finite matrix, too. In other papers we will show algorithms in order to deduce the inverse matrix of an infinite matrix using the extensions to infinite matrices of the QR and LLT matrix factorizations [3] and [4]. ————————–

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Béla Finta / Procedia Manufacturing 32 (2019) 643–646 B´ela Finta / Procedia Manufacturing 00 (2018) 000–000

References [1] B´ela Finta, Analiz˘a numeric˘a, Editura Universit˘a¸tii “Petru Maior” din Tirgu Mures, 200 pages, ISBN 973-7794-14-1, 2004. [2] B´ela Finta, The LU Factorization for Infinite Matrices, Carpathian J. Math., Universitatea de Nord Baia Mare, 2008. [3] B´ela Finta, The QR Factorization for Infinite Matrices, Numerical Analysis and Applied Mathematics, International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2008, Psalidi, Kos, Greece, 16-20 September 2008, AIP Conference Proceedings 1148, ISBN 978-0-7354-0685-8, ISSN 0094-243X, pp. 778-780. [4] B´ela Finta, The LLT Factorization for Infinite Matrices, Numerical Analysis and Applied Mathematics, International Conference on Numerical Analysis and Applied Mathematics, ICNAAM 2008, Psalidi, Kos, Greece, 16-20 September 2008, AIP Conference Proceedings 1148, ISBN 978-0-7354-0685-8, ISSN 0094-243X, pp. 753-755.