LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAICN 510.84 I&63c HO.&1-70- AUG. 51976 Ihe person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN EN tm jaw. ' i APR 8 I fi .'•■ .(a,, •••> a ; x), where (a,, ..., a ) e R is the parameter vector, Y 1 n 1 n and assume that $ depends continuously on the a. as well as on x. For any integer m (0 <_ m <_ n), we then define the family of approximating vectors F = {(f 1 (a,x),f 2 (a,x),..., f (a,x) ): a= (a ir a 12 , .... a lm ,a 2r .... a^, .... a^.a^, ...,-c.J eR , q = m(£-l)+n, and f . (a,x) = = a-,exp(apx) and m = 1. In order to avoid the double-subscript notation, we will henceforth write a = (a, ,a p , ..., a ). The norm used in this paper is the usual uniform norm; that is, if f is any vector- valued function with components f. in C(l), the norm N(f) is defined by N(f) = max ||f. || i 1 where ||f.||=max |f.(x)| . 1 x e I ■ L An element f in F is then called a best approximation to g from F if N(g-f) = inf N(g-f) E p(g) . feP One does not in general expect the existence of such a best approximation since best approximations to a single function from non- linear families often fail to exist. Uniqueness of the best approxima- tion is also the exception rather than the rule, as is the case for the simpler situation when F is a linear subspace [l]. Comments on existence and uniqueness can be provided in particular cases, however, as will be seen in the next section. Our first theorem, analogous to Theorem 85 of Meinardus [8] and having very similar proof, provides a lower bound on p(g). Theorem 1: Suppose that for some a e R and subsets D k C I (k = 1, 2, ..., l) y f (a,x) - g fc U) 4 for all x eD and all k. Suppose also that there exists no parameter b e R such that (1) [g k (x) - f k (a,x)] [f k (b, x ) - f k (a,x)] > for all x e D fc and all k. Ihen (2) p( g ) > inf inf |g,(x) - f,(a,x)|. k xeD k K K Specializing the sets D , one is then led immediately to a sufficient condition for a best approximation. Definition 1: The pair (x,k) is called an extremal of the approximation f (a, . ) to g if |g k (x) - f k (a,x)| = N(g - f(a, .) ). Now let Ej(a) = {x : (x,k) is an extremal} . Theorem 2: If there exists no b e R q such that (l) holds for all x s ^v( a ) an ^ al l k, then f(a,x) is a best approximation to g from F. h In order to obtain a necessary condition for a best approxi- mation, additional hypotheses on F are needed. Henceforth, we assume that the partial derivatives 9f, , s all exist and are continuous for k(a,x) 9a. J (a,x) e R x I. The following theorem then holds. Again, this is a straightforward vector analog of a known theorem [8, p. lUO-l], and the proof will be omitted. Theorem 3: Let f(a,x) be a best approximation to g and assume that p(g) > 0. Then there exists no b = ($ , ..., 3 )e R such that for all k (3) g k (x) - f k (a,x) I 6, 3f k (a > x) j=l " 9a J > for all x e E (a). Notice that both the sufficient and the necessary conditions for a best approximation involve systems of inequalities ((l) and (3), respectively). Computationally, the linear inequalities (3) are much easier to handle than the nonlinear ones (l). It is then of interest to determine whether under some conditions the linear inequalities (3) may also figure in a sufficiency theorem. For approximation of a single function, Krabs [6,7] has discussed several such conditions, one of which, the representation condition [6], readily extends to the vectorial case and yields results applicable to the exponential approximation problem in which we are interested. Definition 2: The family F is said to satisfy the representation # condition if for every pair of functions f(a,x), f(b,x) in F there exist real numbers c.(a,b) (j=l, ..., q) and functions (a,b;x) (k = 1, ..., l) 3 k positive on I such that q . , (k) f (a,x) - f(b,x) = ♦.(a,b;ac) I c.(a,b) 8f k U ' x) for k K K ,1=1 J 3a. k = 1, . .. , £. J The required connection between inequalities (l ) and (3) is then provided by the following theorem, which is an extension of Krabs' Theorem 3 [6] and has a virtually identical proof. Theorem h: Let F satisfy the representation condition and suppose that for some set of closed subsets D. £ I (k = 1, 2, . .., i) there exists no b = (3_, .... $ ) such that (3) holds for all x in D, 1 q k. and all k. Then for all b e R q , (5) min min [g. (x) - f. (a,x)] [f, (b,x) - f (a,x)] <_ 0. k xeLV k k k K IV The conclusion of this theorem is, of course, equivalent to the statement that for no b does (l) hold for all x in D and all k. Putting together the preceding results, we obtain the follow- ing useful characterization theorem. Theorem 5- Let F satisfy the representation condition and suppose that p(g) > 0. Then f(a,x) is a best approximation to g if ana only if there exists no b = (^ ..., Bq ) e ^ such that for all k and all x e E (a) the inequality (3) holds. In applying this characterization theorem, it is useful to keep two things in mind. Firstly, consistency or inconsistency of the inequalities (3) does not depend on the magnitude of the approximation error (N (g - f(a, .))) but only on the signs sgn[g (x) - f (a,x)], which K. K we shall denote by o , (x). Secondly, a necessary and sufficient condition K. for the inconsistency of the set of inequalities (3) (where (x,k) runs over all extremals) is that the origin 6 =(0, ..., 0) of R should lie in the convex hull of the set of q-vectors 9f k (a,x) 3f k (a,x) (6) S = { a (x) ( 5 , ..., ~ ) : (x,k) is an extremal) K oOt dot 1 q. (This result on linear inequalities may be found in Cheney's book [5, p. 19].) This "convex hull" condition yields a set of linear, homoge- neous algebraic equations, from which one may often quickly determine whether or not a given approximation is best. These equations may also be used to obtain alternation theorems [5, p. 75] a fact which will be exploited in the next section. 3. Application to Constrained Exponential Approximation In this section we consider the particular case of approxi- 2 mation "by one-term exponentials (elements of E = {aexp(Bx) : (a,B)£R }), "with the exponential factor B constrained to be the same for all com- ponents. That is, as noted before, we take f = a,exp(a x) for k = 1, ...,£ = q-1; the resulting family of vector-valued approximating functions will be denoted by F . For simplicity, we shall take the interval I exp to be [0, l]. The existence of best approximations from F is readily demonstrated. The compactness of any bounded set { f : feF ; N(f ) < M } exp — is easily deduced from the known compactness result for I = 1 [9]. The usual existence argument then goes through. In order to apply all of the theorems of the last section, in particular Theorem 5, we must first verify that the representation con- dition holds for F . (The smoothness condition prefacing Theorem 3 exp is clearly satisfied.) The representation condition is known to hold for exponential functions[6], but extension from scalar to vector-valued functions is certainly not obvious, because of the requirement that the coefficients c not vary with k. Letting a = (a , ..., a ) and b = (6,, ..., B ), we need (from (h)) (7) a e X p(a x) - B exp(B x) = exp(a x) 4> (a,b;x) {c (a,b) + K. Q JC (^ (^ K K. a c (a,b)x } for k = 1,2, ...» q-1. K q For 4> to be nonvanishing on [0,1 ] as required, the linear factor on the right (in braces) must have a zero at the same point as does the left side. This condition leads to (8) \ i \ - -\ lo «V e k )/<6 q - v (k = 1,2 q -!)• (Equation (8) is obtained under the assumptions that a 6 > (k = 1, • ••, q - l) and 3 - o. ^ 0. The other cases are readily handled by similar arguments. ) Furthermore, the posit ivity of requires also K. that the signs of both sides of (7) should match at any point x . Taking x = 0, ve arrive at the condition (9) sgn (c^ - 3 k ) = sgn (c fe ) (k = 1, ..., q -1 ) . Now since sgn (a - 3, ) = sgn [a log(a ,/3, )], it is clear that by choos- K. K. JtC X K. ing any c such that sgn c =-sgn(3 -a) and then solving (8) for Si M. St St c , . .., c » a set of scalars c , . .., c satisfying (8) and (9) may be found. The functions (a,b) are then defined to be K. a exp(a x) - 3, exp(3 x) \ ~ exp(a q x) { c k + Vq x} , k = 1 , . . . , q - 1, and the representation condition is verified. Then in order to obtain an alternation theorem characterizing best approximations, we may apply the "convex hull" condition as discussed at the end of §2. First note that by Caratheodory 's Theorem [5, p. IT], the condition that the origin 6 of R should be in the convex hull of S may be replaced by the condition that 6 should be a convex linear combination of some q + 1 (or fewer) elements of S. For our exponential approximation, then, the condition is that there exist extremals (x .,k) (with k = l, ..., I, i = l, ..., v , and J v, < q + l) and nonnegative constants X satisfying K. 1 J J X =1 such that .. . ki k l V k (10) ^ A ki °ki eXp(a q X ki ) = ° (k = 1, ..-, q - 1) i=l 4 1-1 \ I 1 A ki a ki a k \± eX P (a q X ki ) = ° * k=l i=l (Here we have used a . to denote a (x . ). ) ki K- ki Of course, only those extremals (x,-> k) for which A is non-zero play a role in actually characterizing a best approximation. Thus one imme- diately sees from (10) that any k for which v = 1 does not enter into the characterization. Considering the various possibilities involving indices k for which v > 1, one quickly arrives at the following K. alternation theorem. Theorem 6: The vector-valued function f is a best approxima- tion from F to g on [0,1 ] if and only if one of the following con- ditions holds. (i) For some index k, f is a best unconstrained approximation K. to g from E • i.e., there are three points of alternation if a, ^ and two points of alternation if a = [8, p. 178] and || g k - f- k \\= N(g-f). (II) There exist two indices (say k = 1,2) with four associated extremals (x^l), (x 12 ,l), (x 21 ,2), (x 22 ,2) such that 0^ ? 0, c* 2 ? X ll < X 12' X 21 < X 22' and °11 °12 = - 1 °21 °22 = - 1 °11 CT 21 = ~ Sgn ^ a i a 2^' Example: Let g = (l,x). The best approximation from F to g on [0,1] is given by f = - e , f = - e gX with 6 = log 2. One readily verifies that N(f-g) = 1/3, extremals are (0,1 ), (l,l), (0,2), (l,2), and the alternation requirements of condition (il) of Theorem 6 are satisfied. 10 Finally, we take up the question of uniqueness. It is obvious that the factors a (k = 1, 2, . .., q - l) are not in general unique. However, it is likely to he the parameter a that is of principal interest, and, as the following theorem shows, this parameter is (with a trivial exception) uniquely determined. Theorem J: Let f(a,x) be a best approximation from F to g. exp Let a = (a , ..., a ). Suppose that either (i) condition (i) of Theorem 6 holds for some k such that a 4 0, or (ii) condition (il) of Theorem 6 holds for some pair of indices k , k such that a ^ 0, a ^ 0. Then 1 d 12 if f(b,x), with b = (3 , ...» B ), is any other best approximation, a = 6 . q. q. The proof of this theorem may be found in detail in [h] . 11 h. Constructi on of Best Simultaneous Exponential Approximations The problem of finding a best approximation to a general g is simplified enormously by the knowledge (from Theorem 6) that one need not consider more than two of the component functions simultaneously. Notice that the key problem is to determine a best a , since then a set of a, 's (k < q) may be determined from the condition that a, exp(a x) k k q should be the best approximation to g from the linear family {aexp(a x) : aeR}. Thus one would proceed by constructing best unconstrained approximations from E to each g . If condition (i) of Theorem 6 is to hold, the best exponential factor must be that for the best unconstrained approximation f such that |f - g | | = max |f - g I I . One then checks to see whether, using this exponential factor as J a the other curves may be fitted to within error If, - g, I I . If q J ' ' k Tc M not, the next step is to examine all pairs g, , g, • For many of these pairs, the best approximation is characterized by condition (I) of Theorem 6 and is therefore of no further interest. If best approximations f , f characterized by condition (il) are then constructed for the k l k 2 remaining pairs, one of these (again the one associated with the largest error) necessarily yields the best a . q Notice that the construction procedure is particularly well- suited to parallel computation. The best unconstrained approximations may be constructed in parallel, following which the prospective a 's obtained from these may be checked for all of the other curves in parallel. Finally, if several pairwise approximations need to be computed, they may also be done in parallel. 12 A FORTRAN program implementing the algorithm has been written with the aid of Dr. Joseph Garber. This program consists of four pieces: A. A main program which does the bookkeeping and makes the comparisons described in the first paragraph of this section. B. A subroutine which, given an exponential factor 3, finds the best approximation aexp(3x) to a single function g. This is the subroutine used to check whether an a obtained from q. approximating a function individually does in fact provide the best overall exponential factor. C. A subroutine which finds the best exponential approximation Sx c e to a single function g. D. A subroutine which computes the best simultaneous approximation to a pair of functions, given that the characterization is by condition (il) of Theorem 6. The functions {g } are handled in tabular form throughout. This is of some importance in analyzing experimental data; some approximation programs (e.g. the University of Illinois subroutine library's Remez algorithm program) require a computable g. This necessitates the introduction of an interpolatory process for tabular data and tends to destroy some of the "minimax" aspect of the approximation. Also, we assume that the functions {g } are essentially positive, in the sense that for each k max g k (x) - min g^ (x) > 0. This eliminates the complicating possibility that the zero function may appear as a best approximation to one of the functions. In the actual 13 coded algorithm and some of the details to follow we have made the even more restrictive assumption that g, >0 for all k. However, no theoretical difficulties should arise in eliminating this restriction so that, for example, "noisy" data which is negative at a few points may be handled. The simplest of Subroutines B, C, D is B. When 3 is fixed, we are dealing with the problem of finding a best approximation from the one-parameter, linear, uni solvent family {aexp(gx): as R}. Such approximations exist, are unique, and are characterized by a two-point alternant. That is, if g is the function to be approximated, then aexp(3x) is the best approximation (on the given interval) if and only if there exist two points x. , x in the interval such that for some constant A g(x 1 ) - aexp(3x ]L ) = A, (11) g(x 2 ) = aexp(6x 2 ) = -A, and p E g - aexp(gx) | = |a|. The single-exchange Remez algorithm, which is known to converge rapidly, is used for Subroutine B. The algorithm proceeds as follows. An initial alternant x , x is guessed. The linear equations (ll) are then solved for a and A. The error p is then computed and compared with |a|. If it is sufficiently close, the procedure is terminated. Otherwise, some point x* at which the maximum error is taken on replaces either x^ or x to form a new alternant x.^ , x , and the process is repeated. The replacement is made according to the rule: x* replaces x if sgn (g(x*) - aexp(3x*)) = sgn (g(x.) - aexp($x )); thus the J J J Ik replacement preserves the alternation in sign. Convergence is very- rapid; usually only one or two iterations are required. Subroutine C is somewhat more complicated but follows along the same lines. In this case we need to construct a best approximation 8x 2 from the two-parameter, nonlinear family {ae : (a, 3) e R }. Characterization is now by three points of alternation; we need to find points x_0 l 2 + g 3 " a t ex P^ x 2 ) + exp(0x )] = 0. Supposing that the parameters a, 3 satisfying these are reasonably close to the a , 3 found initially, we may find corrections Aa , A3 such that (to a good approximation) a = a + Aa , 3 = 3 + A3 by solving the simultaneous linear equations (z 1 + Zg) Aa + a° (x^Z, + X 2 Z 2 > A6 = g l + g 2 " a ° ^ Z l + Z 2^ (z 2 + z ) Aa + a° (x^ + X 3 Z J A ^ = g g + g - a° (z + z ), where z. = exp(3 X.). This is, of course, just one iteration step in the Newton's method solution of the system (lU); but usually this one step is adequate for the determination of a new alternant. (incidentally, the determinant of (15) is just a ^ z 1 z 2 ^ x 2 " x i^ + Z 2 Z 1^ X 3 " X 2^ + Z 1 Z 3 ^ X 3 " x i^» 16 which is always positive; hence (15) may always be solved.) It should "be noted that since Eqs. (12) are not satisfied exactly, the convergence "test in this case does not involve |x| but instead the quantity min (|g(x.) - aexp(3x. ) | } . i = 1,2,3 The validity of this test comes from Theorem 85 in [8], or alternately from the single- function case of Theorem 1. One must, of course, check that the error signs sgn (g(x. ) - aexp(gx.)) alternate on x.^ , x , x . If they do not, additional Newton's method cycles (i.e. the solving of (15) for successive corrections) will be necessary. In numerous tests such additional cycles were never found necessary. Subroutine C as a whole converged extremely rapidly, requiring no more than three iterations (i.e. alternant adjustments) when the given functions {g, } were polynomials. Convergence was just slightly slower, requiring four or five iterations, when the (g, } were exponentials with random noise as described at the end of this section. Two approaches were implemented for the construction of the best simultaneous approximation to a pair of functions g , g . For reference we shall call the resulting subroutines Dl and D2. Subroutine Dl is based on the fact that if (3) has a solution, a better approximation may be constructed from that solution. That is, with any extremal (x, k) of an approximation f(a,x) there is associated a linear inequality (from (3)) of the form (16) a k (x) (3 k + 6 a k x) > 0. If the set of all such linear inequalities (associated with all extremals) 17 has a solution $,,$p>3 . then for some e>0 f(a',x) provides abetter approximation, where a' = (a + eg , a + eB~, a + eg ) . Thus by iteratively searching for extremals, solving inequalities (l6), and correcting the approximation, one may hope to arrive eventually at a best approximation. In trials this method has never failed to converge, although even for simple linear approximation convergence theorems for this type of algorithm do not seem to exist. Convergence is slow, however, and the rate of convergence is highly dependent upon the method of choosing the adjustment parameter e. Thus a complicated e may cut down on the number of iterations required, but the total work involved may not change much. Subroutine Dl was abandoned after trials showed that it required something like four times as many iterations and ten times as much time as Subroutine D2. It is, however, worthwhile to have gained some experience with the straightforward "inequality" method, since it is easily coded and may work in some situations (e.g. for more complicated approximating families) in which a more sophisticated algorithm is impossible. Subroutine D2 is a Remez-type algorithm, very similar to Subroutines B and C. We need to find an f = (cu exp(Bx), a exp((3x)) such that at four points, x^ < x and x < x , the equations g 1 (x 11 ) - a 1 exp(6x i;L ) = A g- L (x 12 ) - a exp(8x ) = - A (IT) g 2 (x 21 ) - a 2 exp(6x 21 ) = - A g 2 (x 22 ) -a 2 exp(gx 22 ) = A hold for A such that |a| = N(g - f). Condition (il) of Theorem 6 will then be satisfied. Elimination of A from pairs of these equations results 18 in the three equations (a) g + g = a [expCBx^) + exp(Bx^ )] (18) .(b) g 21 + g 22 = a 2 [exp(6x 21 ) + exptgx^)] (c) g ±1 + g 21 = a exp(Bx^ ) + a 2 exp(3x 21 ), where here s. . denotes g.(x. t ). B ij B i ij If (l8a) and (l8b) are solved for a , a , respectively, and these values are substituted into (l8c), a single equation in 3 is obtained: £>11 "12 ^21 ^22 (19) H(3) E 1 + exp[6(x 12 - x i;l )] + 1 + exp[6(x 22 - x^)] " g ll " g 21 = ° The algorithm then proceeds as follows. An initial set of points S = {x , x , x , x 22^ is chosen so "that x _ - x^ = x_ 2 - x ? . Equation (19) is in this case easily solved explicitly for the initial estimate of 3, and initial values of a , a , A are subsequently obtained from (l8) and (IT). The usual convergence test — whether |a| is sufficiently close to the maximum deviation N(g - f) of the approximation — is then made, and if it fails the set S must be adjusted. Equation (19) is then solved for a new 3 by Newton's method, values of a , a , A are obtained as above, and we are again ready to test for convergence. It should be noted that (19) has a unique solution, since H'(3) < and H(3) ■+■ ± (g,, + g ?1 ) as 3 ■+ + °°. Adjustment of the proposed alternant S is the most troublesome aspect of the algorithm. By analogy with the usual procedure, we would like the new S (i) to contain a point x* of maximum deviation, and (ii) to be such that the error signs alternate in the desired way at these points. A maximum deviation point x* is therefore located, and an attempt is made to replace one member of S by x* in such a way that (ii) 19 is satisfied. This attempt may fail, for example in the situation sketched in Fie. 1. We see that when we replace l by x* to get the proper sign alternation on x* < x. , the required sign alternations "between curves are not satisfied. To satisfy this requirement, multiple replacements must be made. Thus, looking at Fig. 1, we find that either {x*, x , y,*, x } or (x*, x , x , y *} would be satisfactory as the new set S. In a slightly more complicated case, both x and x may need to be replaced, For example, if g - f is as shown in Fig. 1, but g - f is as shown in Fig. 2, an acceptable new set S would be {x*, x., -, , y-,*» Y 9 *^' 20 Finally, there is the possibility that g - f is something like that shown in Fig. 3. Clearly there is no way in which x* may be inserted into the set S so that the proper sign alternations occur. Looking at Fig. 1, we see that a small shift of t. to the left may have the proper effect of 21 evening out the errors. The program accordingly carries out such a small shift, after first printing out a warning message that an alternant containing x* could not be constructed. This possibility causes a serious difficulty in attempts to analyze the algorithm theoretically; it is, however, not very likely to occur, particularly if the curves fitted are simply "noisy" exponentials. In 33 trails for polynomial g this possibility never did occur, and convergence of the algorithm was very rapid, requiring on the average U.l iterations. Most of the tests of the program were carried out on sets of functions (g v ) which consisted of low-degree polynomials tabulated at 20 equally spaced points on [0, l]. It was felt that such functions would adequately test the main features of the algorithm and give reasonable data on the number of iterations required by the various subroutines, etc. The program as a whole seems to be quite efficient; the best simultaneous approximation to one set of seven polynomial functions was obtained in an "execution time" of under 1.5 seconds on an IBM 360/91. During the course of this computation, Subroutine Dl was called l6 times, Subroutine C 7 times, and Subroutine B ^7 times. Selected excerpts from the output of this test are given in the Appendix. Another type of test is perhaps of more interest from the point of view of practical applications. We artificially generated "experimental" data by adding random errors e (|e| <_. 01) to a set of three exponential curves of the form a e with a = 0.5, 1.0, 1.5. ("Data" points were computed for 20 equally spaced x-values on [0,2].) -h Our program, which identified extremals with a tolerance of 10 , 22 recovered the exponential factor £ = -1 as -1.0000. The traditional way of analyzing exponential data (least -squares straight -line fit to the logarithms of the function values) led to 3 = -0.9988. The difference is largely ascribable to the weighting induced "by taking logarithms; direct least-squares fitting of exponentials is, however, a troublesome nonlinear problem even for a single function. Simultaneous uniform approximation appears to be a very practicable alternative. 23 REFERENCES 1. G. G. Belford, "Uniform approximation of vector-valued functions with a constraint", Math. Comp., v. 26, 1972, pp. ^87-^92. 2. G. G. Belford, "An Algorithm for Fitting Related Sets of Straight- line Data", CAC Document No. 5^, Center for Advanced Computation, University of Illinois at Urb ana-Champaign, 1972. 3. G. G. Belford, "Fitting Related Sets of Straight-line Data". Cybernetics, to appear. h. G. G. Belford, "Vector- valued approximation and its application to fitting exponential decay curves", Math. Comp., to appear. 5. E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966. 6. W. Krabs, "Uber differenzierbare asympototisch Konvexe Functionen- familien bei der nicht-linearen gleichmassigen Approximation", Arch. Rat. Mech. Anal., v. 27, 1967, pp. 275-288. 7. W. Krabs, "Uber die Reichweite des lokalen Kolmogoroff - Kriteriums bei der nicht-linearen gleichmassigen Approximation", J. Approx. Theory, v. 2, 1969, pp. 258-26U. 8. G. Meinardus , Approximation of Functions: Theory and Numerical Methods, English edition, Springer- Verlag, New York, 1967. 9. E. Schmidt, "Zur Kompaktheit bei Exponent ialsuromen", J. Approx. Theory, v. 3, 1970, pp. UU5-U5U. 24 APPENDIX Excerpts from output of test: simultaneous exponential fit to the functions g x = 5 - 3x g 2 = h - 3x g 3 - U - 3x 2 g^ = 5 - 3x g 5 = 7 - 2x gg = 6 - 3x g~ = 7 tabulated at 20 equally-spaced points on [0, 1]. Step 1. Subroutine C is used to find the best exponential approximation to each & in turn. Below are two examples of the output of Subroutine C (identified as "BEANR" in the program). Note that the point of maximum deviation (x* or XMAX) and the points of the proposed alternant (Yl, Y2, Y3) are given both as their actual values and as index values (IMAX, II, 12, 13) locating them in the tabular data. The numbers SIGMAX and NEWSIG actually are only keeping track of the error signs (at XMAX and Yl, respectively). Parameters a, (B are identified as A, B respectively. Computations are for g and g . BERNR SUBROUT I NE CflLi .ED INIT.IRL VRLUE3 - H: 9. 51411D 91 B:— S. 84y~>5D 99 LfiMBDR ''--5IG" 1 : -9. 141060 99 li: i Vi : 6. 88999D 66 12: 19 ¥2: 6. 473680 66 12: 19 Y3 : 6. 94737D 66 ITERATION NUMBER 1 RHO : 6. 198430 86 IMFltf: 26 KMRX : 6. i9988D 81 SIGMAX: 6. i9843D 68 NEW SIG: 6. 141860 66 li: i Vi: 8. 868680 88 12: 18 ¥2 : 8.473680 86 13: 28 V3 : 8.188890 81 A : 8. 5iS14D 91 B : -8. 878320 88 hCVlJ-Qc. '1 > : 8 ±61 371 F v V£ ) ■■ G ( V2 '? ""9 i 61 73[ F 86 88 25 I ! ERA ! I ON NUMBER 2 RHO: @. 161660 86 I MAX: 26 KMflX: 9. 18800D 91 SIGMFlM: 8. 161660 00 NEW SIQ: 0. 14186D ©0 II: 1 VI: 9. 68888D 00 12: 19 V2 : 9. 47368D 99 fl : 8. 516i5D 91 B : ~9. 97044D 80 FCt'D-G^VI.') : 9. 161460 89 h < V2 > -Q ( V2 > : -9. 16146D 98 FdV3?~CKV33 : 9. 1S14SD 99 LAMBDA: 0. 16146D 88 ITERATION NUMBER 3 RHO : 8. 16146D 99 ;:mh : '- 2^ xMfiK 9 i08$gD si 13: 29 V3: 1998£ 91 END OF BEANR SUBROUTINE BEFlNR SUBROUTINE CALLED INI I IAL VALUES - A; 8. 41958D 81 II: 1 VI: 0. 88808D 88 12 B: -8. 119410 81 LAMBDA <-5IG; 18 r'2 : o. 4r"j:6SD 88 ' K 'MA! K - 9 16008D 81 8. 195760 88 12: 18 V2: 9. 47368D 00 ITERA1 ION NUMBER i RHO: 8. 271200 88 IMAM: 28 SIGriliX: 8. 271280 08 NEW SIG II: 1 VI: 0.000O0O OO A: O. 422640 81 B:-8. 123670 k FCt'D-GCt'I? : O. 226390 OO h -G :-8. 22626D 80 F'::V3>-G : 0. 227140 OO LAMBDA: 8. 226260 OO ITERATION NUMBER 2 RHO: 0.227140 OO IMAX: 20 XMAX: 0. 108880 81 SIGMAH: 6. 227140 88 NEW SIG: 0. 19576D "80 II: 1 VI: 0. OOOOOO 98 12: 10 V2 : 8. 4736S0 88 A : 8. 422660 81 B : -8. 1237i0 61 F < VI > -G < VI > : 6. 226640 88 h '.:V2>-G : -8. 226640 86 F(V3>-G : 6. 226640 86 LAMBDA: 6. 226640 86 ITERATION NUMBER 3 RHO : 0. 22664D 88 END OF BEANR SUBROUTINE IMAM: 28 MA .•■' 9. 18800D Ol i: 19 20 13 -0. 195760 00 V3 : 0, 947370 68 8. 10008D Ol O 10O0UO Ol The seven best individual approximations are then tabulated: I Ad) 8(1) 1 8 51615D 61 -8. S78440 88 111 8 422660 01 -8. 1237iD 81 3 8 4*7090 61 -6. 166810 61 4 8 5i-'8950 01 -6. 74*370 66 5 8 764170 61 -8. 334110 66 6 6 612540 81 -6. b .-2890 68 7 6 786600 61 6. 666880 86 RHO(J) 6. 161460 68 6. 226640 66 8. 578880 88 6. 769480 66 8. 416830-81 8. 12538D 66 0. 886880 88 Step 2. For each of the seven exponential factors B(l) determined above, Subroutine B (identified in the program as "BEAGR") 26 is used to determine the best approximation (to each of the other g k in turn) of the form Aexp(B(l)x). Output for B(l) is given below. The maximum error here is identified as "E" in the BEAGR output and "RHO" in the final tabulated output. 1= l BEflQR SUBROUTINE CALLED WITH B=-8. 87844D 60 ITERATION NUMBER i t: 0. 475820 00 LAMBDA:-©. 47582D 00 A: 0. 35242D 01 VI: 0. 00000D 00 END OF BEAGR SUBROUTINE - A: 8.352420 01 E: 0. 47582D 00 BEAGR SUBROUTINE CALLED WITH B=-0. 87044D 00 V2: 0. 10000D 01 iitKHfiUH M'Ji'lb'tK X E : 0. 70311D 08 LAMBDA : -0. 67043D SS A : 0. 355980 01 VI : 0. 31579D 00 V2 ■ R iFiPinnrc ri ITERATION NUMBER 2 t : 8. 68277D 00 LAMBDA : -0. 6S277D 00 A : 0. 40184D 01 VI : 0. 42105D 00 V2 : 0. 10000D 01 END OF BEAGR SUBROUTINE - A: 0. 40i84D 81 E: 8. 68277D 00 BEAGR SUBROUTINE CALLED WITH B=-0. 87844D 00 I itRAl ION NUMBER 1 E ; 0. 11812D 51 LAMBDA : -0. 5S545D 00 A : 0. 8x3120 0i VI ITERATION NUMBER 2 >c "' t : 8. 898540 80 LAMBDA : -8. 82459D 88 A : 8. 58246D 81 VI V2 VI V2 0. 10000D 01 8. 473680 80 0. 08800D 88 8. 578950 88 8. 88888D 80 ITERA1 ION NUMBER 3 E: 8. 870930 88 LAMBDA : -8. 878930 88 A: 8. 587890 81 VI END OF BEAGR SUBROUTINE - A: 8. 5B785D 81 E: 0. S7053D 00 BEAGR SUBROUTINE CALLED WITH B=-8. 87844D 88 ITERATION NUMBER 1 t : 8. 14588D 81 LAMBDA : -8. 14588D 81 A : 8. 84588D 81 VI : 8. 18688D 01 V2: 8. 00088D 88 END OF BtAGR SUBROUTINE - A: 8. 845B8D 81 E: 8, 14588D 81 BEAGR SUBROUTINE CALLED WITH B=-8. 87844D 80 1 ItKHI 1UM MUl'IfcStK X E : 8. 89847D 88 LAMBDA : 8. 11114D 88 A : 8. 68985D 81 VI : 0. 36842D 88 V2 : 8. 18880D 01 ITERATION NUMBER 2 E : S. 43S48D S5 LAMBDA : 8. 34333D 88 A : 8. 634350 81 VI : 8. 00808D 88 V2 ; 8. 10880D 81 ITERATION NUMBER 3 E : 8. 4i287D 88 LAMBDA : 8. 41247D 88 A : 8. 64125D 81 VI : 8. 088880 88 V2: 0. 68421D 00 ITERATION NUMBER 4 E : 0. 41273D 88 LAMBDA : 8. 41273D 88 A : 8. 641270 01 VI : 0. 800000 88 V2: 8. 73684D 88 END OF BEAGR SUBROUTINE - A: y. 64127D 81 E: 5. 412730 00 BEAGR SUBROUTINE CALLED WITH B=-0. 87044D 88 27 ITERATION NUMBER 1 E : 0, 28677D 01 LAMBDA : -8. 28677D 8i fl : 8. 98677D 81 VI : Q. 10000D 91 V2' 00000D 00 END OF BEflQR SUBROUTINE - R; 98677D 01 E: 0. 28677D 61 J R RHOa.. J) 1 8. 51S15D 01 8. 16146D 08 2 0. 33242P 5i 475B2D 88 T^ 0, 48184D 01 8 68277D 88 4 8. 58789D 01 8 87893D 88 5 . 8. 84588D 01 14580D 81 6 8. 64127D 81 8 41273D 88 7 8. 986770 81 28677D 81 Step. 3. A Boolean matrix M = (m ) is constructed with J- J m = T if RHO(l, J) <_ RHO (i) + e (a small tolerance) and m = F i J • ij otherwise. THE COMPARISON MATRIX TFFFFFF (- l F F F F F TTTFFFF T T F II- r F F F F F T F F F F F h F T F F F F F F F T Step, h . If for some i the 1 row of this matrix consists of all T's, B(i) is the best overall exponential factor and the calcination terminates. This did not occur for the example given here, Step 5- Using Subroutine D2 (identified in the output as "PAIRR", the program now computes best simultaneous approximations to pairs {g , g } such that (referring to matrix M) both M and M are p q pq qp F. Printout of results for three such pairs is given below. This is largely self-explanatory, except that (J = 1, K = 2) indicates that it is for the first member (g ) of the pair that the maximum deviation is taken on, and (J = 2, K = l) indicates that it is for the second member (g ). 28 TEP 5 fa » 9*1 PAIRR SUBROUTINE CALLED r . . INITIAL VALUES - ALPHA1: 8. 53299D 61 ALPHA2: 8. 38487D Si BETH : -U. 1B.S5.SD Wl RHO : 8. 36388D 66 LAMBDA . 6. 2651 3D 88 ITERATION NUMBER 1 IMAX' ' : '0 "-'MAX • 8 18888D 61 J: 2 K: 1 bIG: 8. .3638BD 08 111-" I" Xll: 8. 52S32D-81 H2: 18 X12: 6. S3474D 86 121: 2 X21: 8. 5263:2D-6i 122 : 28 X22: 8. i8888D 81_ pji_pj_ipij_ • g 54i53D 81 RLFHR2 : 5. 37887D 81 t ; t i A : -8. 188920 8i LAMBDA ' 6. 27263D 66 RH8: 6. 41535D 88 ITERATluN NUMBEK 2 _ IMAX: 1 XMAX: 6. 66666D 66 J: 1 K: 2 SIQ: 8. 415y5D WW 111- 1 Mil: 8. 88868D 66 112: 18 Xi2: 6. 89474D 86 121- 2 X21: 6. 52632D-81 122: 26 X22: 8. 18888D 61 ALPHAiT 6. 52S81D 61 ALPHA2 : 8. 37682D 61 BE I A : -8. 18713D 81 LAMBDA: 6. 23868D 66 RHO : 6. 41314D 88 ITERATION NUMBER 3 IMAK' 12 XMAX: 6. 57335D 66 J: 1 K: 2 iIG:-8. 4iyl4D 88 111:' 1 Xll: 6. 88888D 66 112: 12 X12: 8. 57895D 68 77-1 • 2 > ; :Z1 ' 6. 52S32P - 8i 122: -26 X22: 6. iSSSSD 81 ALPHA! : 6. 53267D 61 ALPHA2 : 6. 371180 61 BETA : -6. 18286D 61 LAMBDA ' 6. 32668D 66 RHO : 8. 37333D 88 ITERATION NUMBER 4 IMAX: 5 XMAX: 6. 21653D 66 J: 2 K: 1 SIQ: -8. 3799»i> 88 111- 1 Xll: 6. 88888D 68 112: 12 X12: 6. 57895D 88 121- 5 X2i: 6. 21653D 66 122: 26 X22: 8. 16666D 81 ALPHA! : 6. 53365D 61 ALPHA2 - 6. 37721D 61 BETA : -8. 18376D 61 LAMBDA :_ 8. 33643D 66 RHO: 6. 33374D 68 ITERATION NUMBER 5 IMAX: 6 XMAX: 8. 26316D 86 J: 2 K: 1 SIQ :-8. 33974D 88 111: ! Xll: 8. 88888D 68 112: 12 X12: 8. 57895D 66 121: 6 X21 : 6. 26316D 86 122: 26 X22: 8. 18688D 81 ALPHA! : 6. 53371D 61 ALPHA2 : 8. 37761D 61 BE1 A : -8. 18382D 81 LAMBDA: 6. 3371.3D 86 KHO : 6. 33713D 66 END OF PAIRR SUBROUTINE hs > St} FRIRR SUBROUTINE CALLED INITIAL VALUES - RLPHA1 : 5. 35676D 81 RLFHH2: 6. 54878D 61 BETR : -6. 83558D 68 RHO: 6. 16344D 61 LfiMBDR : -8. 252210 5R ITERATION NUMBER 1 IMAX: 12 XMAX: 8. 57535D 86 J: 2 K: I SIG:-8. 163440 81 111: 2 Xll: 8. 526320-61 112: 18 X12 : 8. 83474D 88 I2i: 2 X2i : 6. 52632D-8! 122: 12 X22 : 6. 57835D 86 ALPHA1 : 6. 36321D 81 ALPHA2 : 8. 566880 81 BETA : -8. 62542D 88 LAMBDA: -6. 47726D 66 RHO : 6. 18283D 81 ITERATION NUMBER 2 IMAX: 28 XMAX: 6. 16868D 61 J: 2 K: 1 SIQ: 8. 18283D 81 IMPOSSIBLE TO USE XMAX 111: 2 Xll: 8. 52632D-81 112: 18 X!2: 8. 83474D 86 121: 2 X21: 8. 526320-81 122: 16 X22 : 8. 78347D 66 ALPHA1: 8. 37689D 81 ALPHA2: 8. 55790D 61 BE l A: -6. ,-22980 88 LAMBDA: -8. 37H8D 66 RHO : 6. 82519D 86 29 ITERATION NUMBER 3 I MAX: 25 Xf'iAX : 5. 155550 51 lii: 2 Mil: 8. 52632D-81 121: 2 X21: 8. 526320-81 ALPHA1 : 6. 36010D 81 RLPHR2 : LAMBDA : -8. 54725D 00 RHO : 0. 954850 55 •-' : x c. : d. :=• j. >j : @. BeiSlSu yy 112: 28 X12: 8. 166660 81 122: 16 X22: 8.789470 88 8. 57998D 81 BETR:-8. 84475D 88 ITERATION NUMBER 4 I MAX : 8 XMAX : 0. 363420 5u 111: 3 Xli : 0. 36342D 00 121: 2 K21: 0.526320-01 RLPHR1 : 8. 41263D 01 RLRHR2 : LAMBDA : -8. 649440 88 RHO : 0. 93140D 88 J: 1 K: 2 SIG: -5, 954850 08 112: 28 X12 : 8. 188850 81 122; 16 X22: 8. 789470 88 8. 592830 81 BETA : -8. 91693D 88 ITERATION NUMBER 5 I MAX: li XMAX: 8.578550 88 111: S Mil: 0, 36842D 88 121: 2 X21 : 8. 526320-51 ALPHA!: 8. 369950 81 ALPHA2 : LAMBDA:-©. 712690 86 RHO: 0. 965220 88 •J : ei (<•. : x i>Iu:-5. 5314tf0 85 112: 28 X12: 6. 186880 61 icici: lei Xd'2 : 8. 578950 55 6. 596520 81 BETA: -6. 822660 86 ITERATION NUMBER 6 I MAX: 1 XMAX: 6. 686660 66 Hi: 8 Mil: 6.368420 88 121: 1 X21: 6.866880 66 ALPHA! : 5. 378540 51 RLFHA2 : LAMBDA :-0. 746250 66 RHO : 8. 746250 86 END OF PRIRR SUBROUTINE FAIRR SUBROUTINE CALLED INITIAL VALUES - ALPHA1: 8.318560 81 RHO : 8. 1^.7540 51 LAMBOA : -8. 855280 88 J: 2 K: i SIG: 8.965220 88 112: 28 X12: 8. 186860 61 122: 12 X22: 6. 578950 66 5. 574620 51 BETA : -5. 773680 w [93>9r] ALPHA2: 8.797670 81 BETA : -8. 291520 80 ITERAl ION NUMBER i I MAX: 26 XMAX: 6. i68880 81 111: 2 Xli: 6. 526320-6i 121: 2 X2i : 6. 526.s20~6i ALPHA! : 5. 258520 5! LAMBOA : -5. 156820 5! RHO: 6. 129260 81 J: 1 K: 2 SIG: 6. 137940 81 1 12 : 26 X12 : 8. 166660 61 122: 18 ML^MHii : y. St'^yLJ ox ;94740 86 bt i i-t : — & st-OtZfu otj ITERATION NUMBER 2 I MAX: 26 XMAX : 8.166680 81 111: 2 Xli: 6.526320-61 121: 2 X21 : 5. 526320-5! ALPHA!: 6. 294840 01 ALPHA2 LAMBOA :-8. 11027D 81 RHO : 6. 124760 01 J: 2 K: 1 SIG: -6. 129260 81 112: 28 X12: 8.168880 81 122: 26 X22 : 5. 155550 51 8. 824760 01 BETA: -6. 335350 86 ITERATION NUMBER 3 I MAX: 1 XMAX: 8.868660 86 111: 2 Xli: 8. 526320-61 121: i XS1: 0, tiOOfOO 55 ALPHA! : 8. 292250 51 ALFHA2 : LAriBOA : -5. 111790 81 RHO : 6. 114730 81 J: 2 K: I 12 : 28 1 22 : 25 > ":1!750 51 0. i SIG: 8. 124780 81 X12 : 6. 188880 61 X22 : 6. 186550 61 BETA : -6. 322140 56 ITERAl ION NUMBER 4 I MAX: 4 XMAX: 6. is .''890 86 Hi: 4 Xli: 6. i57890 66 I2i; 1 X21 : 6. 668660 66 ALPHAi : 6. 294480 61 ALPHA2 : LAMBOA: -6. 11277D 61 RHO: 6. 112770 61 J: 1 K: 2 bIG:-6. 114730 01 112: 26 X12: 6. 188680 61 122: 28 X22: 6.186660 61 8. 81277D 61 BETA: -6. 325820 68 END OF PAIRR SUBROUTINE 30 After all pairwise approximations have been computed, a summary of the results are printed out and the pair yielding the largest p = N(g - f) is identified. In this example, the characterizing pair is {gy g }. I J RLPHRKL •J> RLPHR2 i 2 0. 53371D 8i 37761D 01 "0 10382D 01 8. 33713D 08 i £i 9. 442i±D 01 75789D 01 -0 53900D 00 0. D789tiD 88 l 6 0. 48432D 01 62515D 01 -8 76597D 00 8. 25i51D 08 1 7 0. 40088D 01 80900D 01 -0 28768D 00 0. 10000D 81 2 3 0. 32353D 01 77647D 01 -0 60613D 00 0. 76471D 00 2 S 0. 35533D 01 64522D 01 -0 89475D 00 0. 45227D 08 2 7 0. 28947D 01 81852D 01 -0 31845D 00 0. 11053D 01 j 4 Pi. 3F354C 1 01 5 374b2r' 01 — Pi 77j->Z"±l-> 00 0. 74623D 80 3 5 0. "?awM'[n 01 7S468E? 01 - 63£22D 00 0. 84677D 00 3 6 0. 43452D 01 66110D 01 -0 99219D 00 0. 61105D 00 ~, i'" 0. 2944bD 01 Si277D 01 -8 32502D 00 0. 11277D 01 4 5 0. 52187D 01 78028D 01 -0 62008D 00 0. 80285D 00 4 r 0. 42228D 01 80873D 01 -0 31313D 80 8. 10S73D 01 5 6 0. ?4286D 01 55714D 01 -0 48551D 00 0. 42857D 00 a i 0. 64615D 91 75385D 01 -0 i54i5D 80 0. 53846D 00 b "7 0. 58863D 01 P9138D 01 -0 2623SD 88 0. 91305D 00 1= 3 J= Finally, Subroutine B is used to find the best coefficients C in the approximation of the g,'s by C exp (3x), where the B is as obtained above. (in this example, 6 = BETA (3, T) = -0.32502.) Errors p, of these approximations are also printed out. K. i^E hkE DONE. RH8= 0. 11277D 01 BETRa, J>=-0. 32502D 00 K C RH0 1 40638D 01 0. 93616D 00 2 29027D 01 0. 10973D 81 J. V ZShA&u tfl @. lleif' {£> wx 4 42556D yi 0. 18747D 8i 5 69972D 01 0. 55603D- -01 f, 52249D 01 0. 77507D 00 7 81277D 01 0. 1127 i'D 01 UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA -R&D (Sacurlty claaalf teat Ian of tltla, body at abattmct and Incoming armotatlan mual ba antatad whan tha ovarall report It elmaalltad I. ORIGINATING ACTIVITY (Corporal* author) Center for Advanced Computation University of Illinois at Urban a- Champaign Urbana. Illinois 61801 S. REPORT TITLI Simultaneous Fitting of Exponential Decay Curves ia. REPORT SECURITY C t- A isi F I C A T I Of UNCLASSIFIED 26. GROUP 4. DESCRIPTIVE NOTES (Tfma of rapott and tnelualva dataa) Research Report » AUTHOR!*) (Firmt nama, mlddla Initial, la at nama) Geneva G. Be If or d • . REPORT DATE April 1973 7a. TOTAL NO. OF PACES 35 7b. NO. OF REFS 9 Sa. CONTRACT OR GRANT NO. DAHCOU 72-C-OOOl b. PROJECT t*6 c ARPA Order No. 1899 •a. ORIGINATOR'S REPORT NUMRERIS) CAC Document No. 6l OTHER REPORT NOI5I (Any othmr numbara that may ba aaalgnad thla rapart) 10. DISTRIBUTION STATEMENT Copies may be requested from the address given in (l) above. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Research Office - Durham Duke Station, Durham, North Carolina 19. ABSTRACT This paper deals with characterization of best approximations to vector-valued functions. The approximations are themselves vector-valued functions with components depending nonlinearly on the approximation parameters. The .constraint is imposed that certain of the parameters should be identical for all components. An application to exponential approximation is discussed in some detail. DD ,?<,?..1473 UNCLASSIFIED Security Classification UNCLASSIFIED Security Clarification KEY *0«DI ROLE «T Uniform approximation Vector-valued approximation Exponential approximation Nonlinear approximation UNCLASSIFIED Security Classification