■■HHHHi I nnraHl MHKnnnmQraSII HWMffllM WJ U mg HMW M if* ftu»HnBHSIfififl JHHHH KrBffiiB Hot Ha HH IH HB HB8BI mm MSjHIHm msBm MP uu;\; Km ~HHH| HH m Hi Hi ■■HHHfflHiiUfl ■p mm m Wffl nHnflunffi MM BlHilil IMM miH Hja wHagg Brass liMBBtMllBniiffiTOBi llllillill piHHHiil HHyXHi VBBKmXSBKm LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAICN cop- 2. Digitized by the Internet Archive in 2013 http://archive.org/details/methodforsolving521brac Tic. SJ/ Report No. UIUCDCS-R-72-521 TrfAot A METHOD FOR SOLVING POLYNOMIAL EQUATIONS BY CONTINUED FRACTIONS by Amnon Bracha July 1972 DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS THE LIBRARY OE THE S^P 5 1972 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Report No. UIUCDCS^R-72-521 A METHOD FOR SOLVING POLYNOMIAL EQUATIONS BY CONTINUED FRACTIONS by Amnon Bracha Computer Science Department University of Illinois Urbana, Illinois 61801 July 1972 This research was supported in part by the National Science Foundation under Grant No. US NSF GJ 8I3. A METHOD FOR SOLVING POLYNOMIAL EQUATIONS BY CONTINUED FPACTIONS Amnon Bracha Abstract A method for the approximation of all the real roots of an n-order polynomial equation is developed. It is assumed that intervals containing the solutions are known. Bilinear transformations are used to approximate the solution. Convergence is achieved. 1. INTRODUCTION In this paper we generalize earlier results by the same author [1] and develop them for finding all the zeroes of an n-order polynomial equation. In [1] it was shown that for a limited class of functions , such as quadratic or cubic equations, a solution can be approximated by a continued fraction of the form (1.1) J* =— -2 -£ \ *1 + ^ + '•• + \ where A, and B, are determined from the recursion \ and \ (1.2) A. = q. A. . + p. A. „ l H i l-l *i 1-2 B. = q. B. . + p. B. ^ i = 2, 5, l ^i l-l i i-2 with initial values : A = A ± = Pl B = 1 ^ = q The digit set for p. and q. were selected as simple ■binary- constants, e.g., l/2 or 1, in order to reduce the amount of time required to evaluate (1.2). In the current paper we show that polynomial equations of P i any order can be transformed by a bilinear transformation such as x. = 1 V x i+ i into another polynomial equation of the same order, where a simple recursion exists between the coefficients of the two polynomials. The result is that if a method for selecting p. and q. for the 1 ^1 i step can be developed, then we can approximate the solution by using recursion (1.2). The selection method is described in section 5- In section k we develop a method for selecting two constants, a and b for p. and q. and we show the interval of solutions that can be 1 TL approximated by these two constants. By using different pairs of constants we can, therefore, approximate different solutions of the given equation. Theorem 1 in the same section gives a proof of convergence, an important step in the development of the method. Rate of convergence is discussed in detail in [1]. P l P 2 P n q x + q 2 + • • . + V f n+1 A +f ^A . n n+1 n-1 n = 1 2. BILINEAR TRANSFORMATIONS. Following the analysis of Wynn [k] we define a continued fraction as a sequence of bilinear transformations of the form: ( 2 ' 1} f = ?k k -1 2 H k k+1 where f v (x) is a function of x and p , q are constants. The resulting continued fraction is f l = (2.2) A +f ^A . n n+1 n-1 ~ B +f ^ n B . ' n - ±, d, ..., n n+1 n-1 where the functions A. and B. satisfy the recursion (1.2). In a recent paper by the author [1] it was shown, following a note of Wynn [k], that the solution of the Riccati equation (2. If) y' + ay 2 + by + c = , where a, b, c are functions of x or constants, can be expanded as a continued fraction by using a series of bilinear transformations of the form (2.1), and such that each function f^-, k =1, 2, ... also th satisfies the Riccati equation. In [1] the recursions for the k Riccati equation were developed and a method for selecting p, , q, for each step was shown for several functions . In the current paper we develop the results of [1] for polynomials of any order, and we show how the method can be used to find the zeroes of these polynomials. One important assumption in the development of the method in [1] was that p, , q, , are simple "binary constants, i.e. 1 or l/2, and therefore the various recursions that are involved require only "short" operations. In the current paper we generalize this result and include other values for p and q . J. THE SOLUTION OF A POLYNOMIAL AS A CONTINUED FPACTION. We show now how distinct roots of a given polynomial can be approximated by using bilinear transformations. First we develop the recursion for the coefficients of the polynomial for the i+1 l step by using the coefficients of the i step, i = 1, 2, ... . Let n (3.1) P X (x) = 7 ajx^ = be a given polynomial equation of order n. The index 1 refers to the first step in the recursion. In particular, the index 1 is not an exponent, whereas the index k is an exponent when used as a superscript. We use a substitution of the form P i (3-2) x. = — , i = 1, 2, ..., 1 q.+x. , -, t. l+l where p. , q. are constants to be defined. Suppose that the i L step polynomial equation is of the form (3.3) P = a x. + a . x.~" + . . . + a, x. + a_ = , ^ n n 1 n-1 1 1 1 ' then by using substitution (3*2) we have i| 1 a n q.+x. P, \ n V x i+i n-1 / P, \ n " 1 A P^ + . . . + a n — ■ + a^ = q.+x. _ 1 q.+x. , , ^1 l+l *i l+l Multiply by (q. + x. ) to normalize the coefficient of a . We get in i n-1 / \ i , N n-1 i, x n a p. + a n p. (q.+x. . )+...+ a n p. (q.+x. n ) + a^(q.+x. n ) =0 n*! n-1 ^1 VH i i+l y l-^i VH i i+l y VH i i+l ; (3-0 The recursion that follows is i+1 i a = a_ n l+l i i a , = a., p . + na.q. n-1 1 i 0^± n-k i+1 V" k+ t i n-k-t t , * , a k = L Q 1 t ) Vk-t p i q i k -0, 1, ..., n. The resulting i+1 step polynomial equation is ^i+l l+l n , i+l n-1 l+l i+1 P = a x.,-. +a n x . , , + , . . + a, x . , , +a_ n n i+l n-1 i+l 1 i+l The method to approximate the solution of (3«l) can be used now, if an algorithm for selecting p. and q. (i = 1, 2, ...) is defined. This is the subject of section 5. k. INTERVALUES THAT ARE COVERED BY CONTIMJED FRACTIONS In this section we show how to approximate values of the solution by using different pairs of constants p. and q.. Let p. and q. each assume two given values, e.g., p-,q.e{a,b] Continued fractions of the form (l.l) assume all values in the interval [M,m], where M = max lim -j— , . ,. \ m = min lim — — k-*oo Tc A simple analysis shows that M can be found by solving a quadratic equation of the form P M imin mm TKK + M where p , q = max(a.b), and p . , q . = min(a.b) mx Tux 3 ' mm' inm ' Similarly we have for m P. ■mm m = Tnx mx q_ . + m Turn The resulting equations are: q. VT + (a. a + P . - P ) M - p q_ =0 Trim Trim Tnx mm mx mx ttix and nnx %nx Tnin mx " min " min Tnin ~ Assuming that both p and q have the same range, e.g. , p . = q . = a, p =q = b, p ; qe [a,b] then the equation has the form aM 2 + (ab + a - b) M - b 2 = ('4.D P P bm +(ab+b-a)m-a =0 In the following table we give values of a, b and the corresponding ranges [m,M]. a b m M 1/2 1 sfe-l 2 Jk 1 2 fn-} \Tl7-l k 2 ~h -6 i 2 If a solution of the given polynomial equation is known to be in a certain interval, then the appropriate digit set a, b can be used to approximate this solution. 5. SELECTION RULES In this section we develop a method for selecting p. and q. for the case 0 The range of the solution, e.g. , x , can be found now by solving (l|.l). Since we are using a substitution of the form (3.2) our first condition will be: (5.1) m ^ x. g M i = 1, 2, . .. By imposing condition (5-1) we need now only one set of selection rules for p. and q. , i = 1, 2 } . . , , in the range [m,M]. We write below a version of (3.1). Let (5-2) = ^0_ 1 £ 1 k-i E a k x i k=0 where it is assumed that m ^ x, ^ M. We will find p and q such that (5-3) X-, = P l 1 q 1 + X 2 where m ^ x %. M and p , q e{a,b}. Clearly we have four possibilities for selecting p, and q and for each such pair we get different x . In order to make our selection we adopt the inverse approach. We assume that condition (5.1) exists for x , and find the range for x.. for each pair of P 1 and q . We start with the pair p = a, q, = b. From (5-3) we have (5.1+) C = ^- ^ x, i-^-=c b+m 1 b+M 10 Since a, b, m and M are known, C and c can be found and we have defined a range for x for which a selection of p = a and q = b will assure the condition (5.1)- Since x is an unknown we substitute the results of (5«0 in (5-2) in order to find the allowable range for p = a and q = b. We have 1 - a o c g ^ C £ 1 k-1 E a k x i k=0 and this result is possible for any x, in the range [c,C], therefore we conclude that if the following two conditions are satisfied (5-5) n E 1 k a. C k > k=0 n E 1 k a n c k < k=0 we select p-. = a and q = b. The analysis can be carried now for each of the remaining three pairs of values of p and q. Since x p has the same range as x, and satisfies i polynomial of the same degree, we can use the same procedure for x , etc. The result is that the entire range is divided into four sections. For each section we can choose a pair of p. and q. , such that condition (5-1) will be satisfied for x. ,. Clearly, by using only the upper bound in (5-5); f° r each pair of p. and q. we can define a unique set of selection rules. Our next objective is to show that there is an overlapping between two consecutive regions so that by using only the upper bound for each pair, the entire region is covered. 11 It can be verified that the four regions defined by (5*5) for each pair p. and q. are: (5.6) (a) p = K, q = a (b) P = b, q =b (c) p = a, q = a (a) p = a, q -D. . In the theorem which follows we give a necessary and sufficient condition, for the overlapping of the regions defined in (5.6), for two cases. The analysis for other values of a and b is similar and therefore is omitted. Theorem 1 : The regions defined by (5.5), for each of the pairs (5-6) overlap each other if and only if the following conditions are satisfied: (a) If < a < b, then M • m g 1/2 . (b) If b < a < 0, then M • m 1 1/2 . Proof : For < a < b the regions defined in (5-6) are in decreasing order. Therefore we only have to show that the upper bound for the pair (5.6)-(b), (c), (d) is greater or equal to the lower bound of (5.6)-(a), (b) , (c) respectively. For the upper bound of (5-6)-(b) and lower bound of (5. 6)- (a), we have, (5.7) b > b w u b+m ~ a+M ' for the second pair we have, a b a+m b+M 12 and for the third (5.8) b+m a+M The conditions that must be satisfied are : (5. 9 ) M - m i? b - a M • a ^ m • b From the definition of a,b, the range [m,M] and condition (5.1) we have : b ,, -, a — — = M and r— — = m . a+m b+M Eliminating a and b we get mM(l+m) a = 1-mM and mM(l+M) 1-mM The first condition in (5-9) is satisfied if and only if ^ , mM(l+M-m-l) mM(M-m) M-m§b-a= ^ — - — = ^ — r-r— 1-mM 1-mM or mM I 1/2 . For the second condition we have, M 2 m(l+m) > m 2 M(l+M) 1-mM 1-mM or M 1 m which was assumed. For the second case, b < a < 0, the inequalities in (5-7) and (5.8) reverse and therefore the result follows. 13 6. CONVERGENCE CONDITIONS In this section we develop necessary and sufficient conditions for the procedure to converge. Theorem 2 : Let .. A P l V _2 P n ( } X i^ B% 1+q2+ ... V x n+1 be a continued fraction representation of a solution of (3.1) which was found by substitution (5-2). p.,q.e{a,b}, x. e [m,M], {±=1,2,. . .) , where relations (l|.l) exist between a, b, m and M. Necessary and sufficient conditions for (6.1) to converge to a solution of the polynomial equation (3«l) are: (a) ^ (2m+a) > (b) 2 + ^ (a+m-M) > Proof: Let A /B be the n approximation to x, . We will show that n' n 1 under condition (a) and (b), for every e > 0, there exists N ? such that for all n > N A n ^ 5 n = B - B" < £ > n or equivalently that if T = 5/8 p n =3A> ••• then under the conditions of the theorem It I < 1. 1 n' It was already shown in [lj that 1 - — x P n T = 2 % B n-1 1 + P B „ n n-2 Ik therefore the condition It I < 1 implies: 1 n 1 i _!h x < i + ^J^i P n p B _ n *n n-2 and . q n n-1 ^ . Ti -1 - - < 1 X P B _ p n n n-2 n For the first expression we have: - (x + 3a=t ) > o , P n n B n-2 and since this condition is true for every x it can be replaced "by m, ■p also min n ~-^ = a+m and the result is: - (a+2m) > . P For the second expression we have q B . n / n-1 v - „ 2 + — ( = - x ) > P B „ n' ^n n-2 B n-1 here we substitute M for x and the minimum value of _ , and the second n B _ ' n-2 condition follows: 2 + - (a+m-M) > . P 15 7. CONCLUSIONS A method for the approximation of real roots of an n-order polynomial equation is described. The assumption is that intervals, each containing one solution are known. Those intervals are later used according to the analysis of section k as m and M in order to define a pair of digits a and b, which will be used in a bilinear transformation such as (3-2) to approximate the solution. The algorithm described in this paper consists of the steps (1) Select a digit set, a and b, which covers one solution at a time. (2) Use iteration (2. 3) to approximate the solution. (3) Iterate on (3«'+) to get the coefficients of the i+l^k polynomial from the ith polynomial equation. (k) Use selection rules, such as (5.5), which covers the entire allowable region, for the next p . and q . . (5) Check for accuracy and if obtained divide A, by B to receive one solution. (6) Repeat (l) to (5) for each interval which contains a zero of the given polynomial equation. Notes (a) If the condition of theorem 1 is not satisfied for a certain interval, this interval can be divided into subintervals. (b) By using powers of 2 for pj_ and q^, the time required to evaluate most of the expressions is reduced if a binary computer is used. (c) An example of a polynomial equation of order 5 is given in appendix A. 16 REFERENCES [1] A. Bracha, "Application of Continued Fractions for Fast Evaluation of Certain Functions on a Digital Computer/' presented in the IEEE Symposium on Computer Arithmetic , University of Maryland, May 1972. Also Department of Computer Science, University of Illinois, Report No. 510, March 1972. [2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford 1954. [3] H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc. , New York, 1948. [4] P. Wynn, "On Some Recent Development in the Theory and Application of Continued Fractions," SIAM Journal on Numerical Analysis Vol. 1, pp. 177-197. 1964. 17 APPENDIX A 18 We show now how to approximate one solution of a given polynomial equation of order 5: Let ' {x) = ix+)jix+ - I3x 5 + ljx 2 + 36x - 36 = P 5 (x) = (x+3)(x+2)(x-l)(x-2)(x-3) 5 k = x - x be a given polynomial equation of order 5, and suppose that it is known that the interval s/17-3 s[rj-± k ' 2 contains one solution. Our first step is to select a pair of digits _, a and b , according to the analysis of section k, such that every value in the given interval can be approximated by a continued fraction, p,qe {a,b). From the table at the end of the section we have: a = 1, b = 2. The recursion relation between the coefficients of the i and i+2_th polynomials of order 5 are : i+1 i a 5 ~ a o \ = a i 5a o q i i+1 i 2 , , i |1A i2 a^ « Vi + ^i*L +10a O q I i+1 i 3 J7 i 2 . /- i 2, nn i 3 a_ = a,p. +3a p.q. +6a,p.q. + 10 a_ q. 2 31 2 1 1 1 1^1 n i i+1 i h _l „ i 3 , i 2 2 . , i 5,. H a l = Vi 2a 3 P q i + 5a 2 P i q i + k a l p i q i + 5a q i i+1 i 5 ilf x i 32 x i23 x i L i 5 a„ = a._p. + a. p. q.+ a_p.q. + a^p.q. + a,p. q. + a~q. fi k in 3 x * 2 1 1 I11 O^-i 19 The selection rules according to the analysis of section 5 , are: for P for P" [5->/l7 2 i z o, p i = x > q i = 2; fv/TF-il § o, n — 1 n — 1 • 1 * 1 Pj_ - -L* CL = 1, for P^ (5-^17) ^ 0, P i =2, q. a 2; Otherwise p. =2, q. = 1 20 k P k q k \ \ V\ Error 1 2.C l.C C.C C.1CCCD CI C.C C.10000C 01 2 2.0 l.G G.200CC CI 0.1000C 01 C .2CCCCCCCCCCCCCCCC CI -C.1C00CC CI 3 2.C l.C _C.2CCCC CI 0.3CCCC_C1 C . 6666 6 66666666666 C PC C.22222C CC 4 2.0 1.0 C.60CCC CI C.5CCCD CI C. 12CCCCCCCCCCCCCC0 ~C1 -C.2C0CCC 00 5 2.C l.C C.1CCCC C2 0.1100C 02 C .9C9C9C9C9C90909 1C OC C.9C9C9C-C1 6 2.C l.C C.22CCC C2 C.21CCC C2 C. 1C4 76 190476 19048C 01 -C.47619C-C1 7 2.0 l.C C.42CCC C2 C.42CCC C2 C . 9 76744 1 E 6C465 1 160 CC C.23256C-C1 8 2.C l.C C.P6CCf: C2 C.85CCC 02 C.1C1 17647C58E2352C CI -C.11765C-C1 9 2.0 l.C C_.17CCC_C3 C.171CC C3 C. 994 152C467 E36257C CC C.58480C-02 1C 2.6 l.C C.342CC C3 '0.34100 C3 C.1CC29 2255 121964 EC C 1 -C.29326D-C2 11 2.C l.C C.662CD C3 C.682CC 03 C. 99 E53587 11 5666 16C CC C.14641C-C2 12 2.C l.C C.1366C C<« 0.1365C C4 C. 1CCC7226CC 1226 CI C CI -C.7326CC-03 13 2.C l.C C. 27300 04 0.2721C C4 C . 9996 2 28 2 276 C52 7 2C CC C.266170-C3 14 2.C l.C C.5462D C4 C.5461C C4 C . 1CCC 183 1 166453C3C CI -0.1E212C-C3 15 2.0 l.C_ C.1C92C C5 C.1C92D C5 C . 99 9 9C84 5C C595C 75C CC C.9155CC-C4 16 2.0 l.C C.2ie5D C5 0.2185C C5 C .1CCCC45 777C6569CC 01 -C.45777D-C4 17 2.0 l.C_ C._4269D C5 C.4369D C5 C. 9999771 1 199102 79C CC C.22e88C-C4 18 2.0 l.C C.£738C C5 C.e738D C5 C . 1CCCC 1 1 44* 125452C CI -C.11444C-C4 19 2.C l.C C.174EC C6 C.1748C 06 .9999942 7 7965C 15 5C CC C.5722CC-C5 2C 2.C l.C C.3495C C6 C. 24950 C6 C . 1CCCC0286 1025678C CI -C.2861CC-C5 21 2.0 l.C C.699 1C C6 0_.69 9_1C 06 C . 999 99 E 56 94 E92C 7 5 CC C.143C5C-C5 22 2.C l.C C.1398D C7 0.1398D C7 C. 1 CCOOOO 7 152559C8C 01 -C . 71526C-06 23 2.C l.C C. 27960 C7 0.2796C C7 C. 9 999996423 72174 CC CC 0.35763C-06 24 2.C l.C C.5592C C7 0.5592C C7 C . 1CCC0CC 178 E 12945C CI -C.178eiD-C6 25 2.C l.C C. 1 1 180 Cft 0.U1EC CE C . 99999991C5930355C OC C.e94C7D-C7 26 2.0 1.0 C.2227C C8 C.2227D CE C. 1CCCCCCC447C3484D CI -C.447C3C-07 27 2.0 l.C 0.4474C C8 0.4474C 08 C .99999997 764e2583C CC C.222520-C7 28 2.0 l.C C.E94eC C8 0.E94EC C8 C. 1CCCOO00 11 17587 1C 01 -C.11176C-C7 29 2.0 l.C C.179CC C9 C.179CD C9_ _C?_9595999944 12C645C CC _ C.55879C-C8 3C 2.0 l.C C.3579C C9 C.3579C C9 C. lCCCC000C279296eC CI -C.2794CD-08 31 2.C hC C_.7156C C9 C.7156DjC9 C. 99999999860_20 16_1C_0C _ C.13970C-08 32 2.0 1.0 0.1432C 10 0.14320 10 C. 1CCCCC0CCC69E492D CI -C.69e49D-09 33 2.C l.C C.2E6 30 10 0.28630 10 0. 99999 9 99965C754CC CC 0.3 49250-C9 34 2.C l.C C.5727C 1C 0.5727C 1C C. 1CCCCCCCCC 1 74623C CI -0.17462C-09 35 2.C l.C 0.1145C 11 0_.1145C 11 C. 99_9 9999999 126J85_0 _CC _ C. 873110-10 36 2.C l.C C.2291C 11 C.2291D 11 C . 1 CCCCCCCCCC43656C CI -C.43656C-1C 37 2.0 1.0 C.45_E1C 11 0_.45eiD 11 C. 9999999999 78172 10 O C C.21826C-1C 38 2.C l.C C. 91630 11 0.9163C 11 C .1CCCCC00COC 109 14C CI -C. 1C9 14D-1C 39 2.C l.C C.1E33D 12 C.1E33D 12 C. 999999999994543CC OC 0.5457CC-11 40 41 2.0 2.C 2.C 2.0 2.C 2.C l.C l.C l.C l.C l.C l.C C.2665C 0.733CD C.1466C 0.2932D C.5E64C C.1173C 12 12 12 13 13 14 C.2665C 0.7330C C. 14660 0.2932C 0.5864D C.1173C 12 12 C.1CCCCCCCCCCC272EC 0.9999999999986257C CI CC -C.27285C-11 C. 126420-11 -C.6e212C-i2 C.241C6C-12 42 43 13 13 13 14 C.1CCCCCCCCCCCC662C C.9999999999996589C CI CC 44 45 C.1CCCCC0000000170C C.9999999999999147C CI CC -C.17C530-12 0.85265C-13 46 47 2.C 2.C 2.0 2.C 2.0 l.C l.C l.C l.C l.C 0.2346C C.4691C C.9382D 0.1876C C.2752D 14 14 14 15 15 0.2346C C.4691C C.92E2D 0.1876C C. 27530 14 14 14 15 15 C.1CCCCCCCCCCCCC43C C.9999999999999787C CI OC -C. 426330-13 0.21316C-13 48 49 50 C.1CCCCCCCCCCCCC1CC C.9999999999999947C C.1CCCCC00CCC000C3C CI OC 01 -0.1C658C-13 C.52291C-14 -C.26645C-14 IOGRAPHIC DATA !T 1. Report No. UIUCDCS-R-72-521 3. Recipient's Accession No. 5. Report Date July 1972 le and Subtitle A METHOD FOR SOLVING POLYNOMIAL EQUATIONS BY CONTINUED FRACTIONS thor(s) Amnon Bracha 8- Performing Organization Rept. No. rforming Organization Name and Address Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 61801 10. Project/Task/Work Unit No. 11. Contract /Grant No. US NSF GJ 8I3 )onsoring Organization Name and Address National Science Foundation Washington, D.C. 13. Type of Report & Period Covered Research 14. jpplemenrary Notes A method for the approximation of all the real roots of an n-order polynomial equation is developed. It is assumed that intervals containing the solutions are known. Bilinear transformations are used to approximate the solution. Convergence is achieved. yW ords and Document Analysis. 17a. Descriptors continued fractions , Riccati equation, bilinear transformation •ntifiers 'Open-Ended Terms I SATI Field/Group ► lability Statement Release unlimited ' f IS-35 (10-70) 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 22 22. Price USCOMM-DC 40329-P7I SEP 2 1 197? EH I ■ rnn? no W ' 0£M in iH i a ■I ■ Hi • ■ i .>» *-jt I I I m m m ftttttfimIGS ■HO ■ m m H ■ H