P. R. Schlegel. August oc\ CIRCULATING COPY REPORT NO CHARACTERIZATION OF SOLUTIONS OF THE EQUATION e Ax x.

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1 oc\ " -, «,,, CO REPORT NO. 192 CIRCULATING COPY CHARACTERIZATION OF SOLUTIONS OF THE EQUATION e Ax x. P. R. Schlegel August 1976 Approved for public release; distribution unlimited. USA BALLISTIC RESEARCH LABORATORIES ABERDEEN PROVING GROUND, MARYLAND

2 Destroy this report when it is no longer needed. Do not return it to the originator. Secondary distribution of this report by originating or sponsoring activity is prohibited. Additional copies of this report may be obtained from the National Technical Information Service, U.S. Department of Commerce, Springfield, Virginia The findings in this report are not to be construed as an official Department of the Army position, unless so designated by other authorized documents.

3 SECURITY CLASSIFICATION OF THIS PAGE (When Dele Entered) 1. REPORT NUMBER REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER REPORT NO TITLE (and Subtitle) Characterization of Solutions of the Equation e Ax = x. 5. TYPE OF REPORT 4, PERIOD COVERED Final 6. PERFORMING ORG. REPORT NUMBER 7. AUTHORfoJ 8. CONTRACT OR GRANT NUMBERfaJ P. R. Schlegel 9. PERFORMING ORGANIZATION NAME AND ADDRESS U.S. Army Ballistic Research Laboratories Aberdeen Proving Ground, MD PROGRAM ELEMENT. PROJECT. TASK AREA A WORK UNIT NUMBERS RDTE Proj. No. 1W16112AH43 II. CONTROLLING OFFICE NAME AND ADDRESS USArmy Materiel Development Readiness Command 51 Eisenhower Avenue Alexandia, VA REPORT DATE AUGUST NUMBER OF PAGES U. MONITORING AGENCY NAME ft ADDRESSfif different from Controlling Ofllce) 15. SECURITY CLASS, (ol this report) 21 Unclassified ISa. DECLASSIFI CATION/DOWN GRADING SCHEDULE 16. DISTRIBUTION STATEMENT (ol thle Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (ot the abmtract entered In Block 2, II dltlerent from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reveree elde II necaaeary end Identity by block number) Characterize, Solution, Rolle's Theorem ABSTRACT fcanf nue era reverm* *fa> ft nejceaaary and. Identify by block number) This paper characterizes the solutions of the equation e Ax any real number. = x, where A is EDITION OF I NOV 65 IS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Wrten Dele Entered)

4 TABLE OF CONTENTS Page I. INTRODUCTION 5 II. CHARACTERIZATION OF SOLUTIONS 5 III. CONCLUSION 9 ACKNOWLEDGEMENT 19 DISTRIBUTION LIST 21

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6 I. INTRODUCTION This paper characterizes the solutions of the equation x = e^, (1) which arises in several areas of investigation, e. g., determining the maximum range of an artillary trajectory for fixed time under a given aerodynamic drag [1]; determining the residual velocity of a bullet traveling through a liquid as a function of fuze delay time and striking velocity [2]; calculation of the burning rate for constant frequency in a T-Bumer experiment [3]; and in the problem of intense surface heating of a slab [4]. II. CHARACTERIZATION OF SOLUTIONS We will now state and prove, a series of lemmas, which will characterize the solutions of e Ax = x as a function of A. The first three lemmas will give the algorithms for generating the solutions. In the sequel a value of x which satisfies e Ax = x will be referred to as i solution. Also, log(x) will denote the natural logarithm of x. -1 Ay Lemma 1. For < A < e, the sequence y.. = e 'i, y = e, is a bounded monotone decreasing sequence and its limit is a solution. Ay Proof: y f =e>e / =y,. Assume y~ > y. >... > y.. Then McCoy, H., Private consultation. Walther, R., et al, Forthcoming report concerning vulnerability analysis of MiG-21 aircraft to 2 mm and improved 2 mm projectiles with various types fuzing. 3 Ibiricu, M. M., Forthcoming report on analysis of T-Burner results based on A-13 propellant data. Masters, J. I., "Problems of Intense Surface Heating of a Slab Accompanied by Change Phase," Journal of Applied Physics, Vol. 21, 1956, pp

7 .. = e ^i < e ^i-1 = y.. Hence, by induction the sequence is monotone decreasing and bounded by zero; in fact, it is bounded below by e. Therefore, y. ->- y and y is a solution. Lemma 2. For < A < e, the sequence y. 1 = A log(y,)» y = A, is a bounded monotone increasing sequence and its limit is a solution. Proof: For < A < e" 1, y : = A" 1 logca~ 1 ) > A" 1 log(e) = A" 1 = y Q.... Assume y n < y, <... < y., then y., = A" log(y.) > A" log(y. ' ).= y.. u i i i+i i i-j. i Thus, by induction the sequence is monotone increasing. To show the sequence is bounded, we will generate a bound for the first three terms, assume a form of the bound on y,, k = 1,..., i, then -1 use induction. To simplify notation, let U = log(a ). Then y 1 = A" 1 U, y_ z = A log(y i ) = A [U + log(u)], y 3 = A^logCy^ = A'V + log(u + log(u))] = A" X [U + log(u) + log(l + U~ 1 logcu))]. '. Since log(l + x) < x for x >, then y < A~ 1 [U + log(u)[l + U" 1 ]]. Note that U > 1, this implies log(u) J>. Assume the form of the bound holds for y., that is, y. < A" 1^ + log(u)[l + U" U 2 " 1 ]]. Then L-l = A log(u)[l + U" U 2 " 1 ]])]

8 < A'V + log(u + log(u)[l + U" U 2 " 1 ])] < A -1 [U + log(u) + log(l + logcuhu' U 1 " 1 ])] < A -1 [U + log(u)[l + U" U 1 " 1 ]]. Thus, the same form of the bound holds for y. Now k= k= then for all i y. < A" X [U + U(U - l) Therefore, y. -* y which satisfies -1 A log(y) or y = Av e, that is, y is a solution. Ay Lemma 5. For A <, the sequence y., = e 7 i, y = e, has a unique limit and its limit is a solution. Proof: By induction we will show the terms of even subscripts form a bounded monotone decreasing subsequence and the terms of odd subscripts form a bounded monotone increasing subsequence. Now that is, and Ae Y 1 = e < e = y y Ay, Ay.. 2 = e 7 1 > e 7 = y l> Y 1 < Y 2 < e = y, Assume for i even

9 Then y i _ 1 = exp(ay i _ 2 ) < expfay.^ = / i+1 = exp(ay i ) For i odd assume Then y^ = exp(ay i _ 1 ) < expcay.^ = y i+1 = expcay.^ < exp(a yi _ 2 ) = y^. Hence, by induction the sequence has the following ordering: y l < y 3 < y 5 < ' < y 2i+l «' < /2i < ' ^4^2" V Suppose y_. 1 -» y_ and y. -* y p, where y~ and y p are distinct. Then by Rolle's theorem, there exists x 1 e (y n,y p ) such that -r (e - x) _' J. U C ujc JV~*A_ " \J This implies expfax^ = A <. This is a contradiction. Lemma 4. For A < there exists a unique solution. Proof: This follows immediately from lemma 3 and Rolle's theorem. Lemma 5. For < A < e there exist exactly two solutions. Proof: From lemma 1 and lemma 2 there exist two distinct Ax solutions, call them x.. and x?. Let h(x) = e - x. Suppose there exists a solution x distinct from x and x. Now -j = Ae - 1, that o J., dx is, -T vanishes only at x = A log(a ). This is a contradiction, since Rolle's theorem says -j must vanish for at least two distinct points. Lemma 6. For A > e there exist no solutions. AY Proof: Let g(x) = e and f(x) = x. Now g(x) > 1 + g'()x and g' () > e. Then for a point, x, of intersection (a solution) of f(x) and g(x), X Q > For x > 1.58, g(x) > g(1.58) + g'(1.58)(x ). This implies x > 4.86 > e. Since g(e) > e and since, for x > e, 8

10 g'(x) > g'o) > 1, there exist no solutions. -1 Ax Lemma 7. For A = O o r A = e,e =x has a unique solution. Proof: Trivially, A = has the unique solution x = 1. For A=e,x = e i s a solution. By assuming another solution x.. ^ e, this results in an immediate contradiction to Rolle's theorem. III. CONCLUSION The previous seven lemmas have characterized the solutions of e Ax = x. Figure 1 gives a graph of 1the solutions, and a tabular form of the solutions is given in Table 1. There are more general forms of equations for which solutions are desired. By an appropriate change of variable, these can be readily Ax reduced to the form e = x. For example, consider the solutions of x + a=be c(x+d). (2) Let z = (2L± r i) e c C a - d ) or x = e ccd " a) bz - a, then C2) reduces to z = e z, where A = bee ^ '. Another form is Q ax = exp(bx ), (3) then x = a z, where z is a solution of e = z for A = bca For A <, lemma 4 shows the existence of a unique solution. Lemma 3 gives a simple algorithm for approximating the solution, but for A < -1, this algorithm converges slowly. Fritsch, Shafer and Crowley [5] considered the case for A < and gave an efficient algorithm for approximating the solution. 5 Fritsch, F. N., Shafer, R. E. and Crowly, W. P., "Solution of Transcendental Equation we W = x", Communication of the ACM, Vol. 16, 1973, pp

11 e Figure 1. Solution of X = e AX 1

12 TABLE 1. SOLUTIONS OF THE EQUATION X = e AX r C "J '6 \ ?.79 C' 5. C b 5 9 ie t459

13 TABLE 1. SOLUTIONS OF THE EQUATION X = AY e (CONTINUED) L Iff CO 44.7G

14 TABLE 1. SOLUTIONS OF THE EQUATION X = e AY (CONTINUED) OC3.6? ' C O.OS3.5? ? (i 3X CO C f' (

15 AX TABLE 1. SOLUTIONS OF-THE EQUATION X = e (CONTINUED).3 Q S

16 TABLE 1. SOLUTIONS OF THE EQUATION X = e AX (CONTINUED) A , X X 2 15

17 AY TABLE 1. SOLUTIONS OF THE EQUATION X = e (CONTINUED) A X l C X 2 16

18 AY TABLE 1. SOLUTIONS OF THE EQUATION X = e (CONTINUED) A ' 17.5 X,

19 AY TABLE 1. SOLUTIONS OF THE EQUATION X = e (CONTINUED) A X l X

20 ACKNOWLEDGEMENT The author gratefully acknowledges Dr. M. S. Taylor for his many useful editorial comments which were incorporated in this paper. 19

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22 DISTRIBUTION LIST No. of Copies Organization No. of Copies Organization 12 Commander Defense Documentation Center ATTN: DDC-TCA Cameron Station Alexandria, VA Commander US Army Materiel Development and Readiness Command ATTN: DRCDMA-ST 51 Eisenhower Avenue Alexandria, VA Commander US Army Aviation Systems Command ATTN: DRSAV-E 12th and Spruce Streets St. Louis, MO Director US Army Air Mobility Research and Development Laboratory Ames Research Center Moffett Field, CA Commander US Army Electronics Command ATTN: DRSEL-RD Fort Monmouth, NJ Commander US Army Missile Command ATTN: DRSMI-R Redstone Arsenal, AL Commander US Army Tank Automotive Development Command ATTN: DRDTA-RWL Warren, MI Commander US Army Mobility Equipment Research Development Command ATTN: Tech Docu Cen, Bldg. 315 DRSME-RZT Fort Belvoir, VA Commander US Army Armament Command Rock Island, IL Commander US Army Harry Diamond Labs ATTN: DRXDO-TI 28 Powder Mill Road Adelphi, MD Director US Army TRADOC Systems Analysis Activity ATTN: ATAA-SA White Sands Missile Range NM 882 Aberdeen Proving Ground Marine Corps Ln Ofc Dir, USAMSAA 21