A Set of Test Matrices

Transcription

1 a set of test matrices 53 A Set of Test Matrices. Introduction. For the testing of the adequacy and efficiency of any computing method it is clearly very desirable to have readily available a set of reliable test data. The material discussed below has been found by the author to be of great utility for this purpose. It is surmised that it may also be of interest to many others engaged in the testing of new techniques for the inversion of matrices and the determination of characteristic roots and vectors. Furthermore, the results may be of interest also for purely theoretical reasons. The exact inverses of a sequence of ten nonsymmetric and again of ten symmetric matrices of extremely poor condition, their determinants, and their largest and smallest characteristic roots and associated vectors as well have all been calculated. These are available in the UMT file. These matrices were obtained by a simple modification of the well known Hilbert matrix, and while they share many features characteristic of that matrix, they differ from it in many other important aspects. The data calculated also permit an examination of the closeness of certain bounds proposed by Ostrowski, Parker, et al. 2. The sequence of matrices. The set of matrices mentioned above is obtained from the sequence of "positive" matrices. 2-3" 3-4- i (n + l)- (» + 2)- n =, 2, 3, [ trl in + l)- (2«- l)-. Thusaij = for alii =, 2, 3, and Oij = (i + j )_ for i = 2, 3, j =,2,3, The matrix An has the following properties: a. The determinant dn of An is of the form ( )"-5 -, where 5 > 0 is an integer. b. The elements of An~l are integers. Further, for j = 0 for j t±. c. The characteristic root of largest absolute value is positive, that of smallest absolute value negative. The nonsymmetric matrices A are thus finite segments of a matrix A which is closely related to the well known symmetric Hilbert matrix (2.) B = (jb«), bn - + )- for i, j =, 2, 3,. The matrix B has been discussed in many places, e.g., by Todd [], and many of its properties are now known.

2 54 A SET OF TEST MATRICES In giving brief sketches in substantiation of some of the properties of An we shall prove first that 8n satisfies the following recursion formula: (2.2) for n =, 2, 3,, with 8t =. Multiplying each column of dn+ by the denominator in the last row, and then dividing outside by the product of these factors, yields (n + l)(n + 2)---(2n + ) n + n + 2 n + n + 2 2n + 2n + n + 2 The following further steps, carried out in the order indicated, lead to (2.2):. Subtract the last column from each of the others, making use of the fact that n + j 2n + in - i + ) («- j + ) i + j n + i (i + j - l)(n + t) 2. Remove from the first n columns the respective factors n, n, to the outside, and reduce the determinant to the order n. 3. Finally remove from all the rows except the first the respective factors getting (2.3) dn+ = n n n + 2 ' n + 3 ' ' 2n' n\(n )! (n + )(» + 2) (2w + l){n + 2)(«+ 3) 2re L (n + 2) (2n) J ' (2» + l)(n + ) _( 2n V2 (2n +!)(«+ ) which proves (2.2). The solution of the difference formula (2.3) leads to (2.4) dn = (-)"- for n =,2,3, Consequently, [!2! (» - 2)!(n - l)!]3 (m - l)!(w +!)!(«+ 2)! (2m - )!, x [!2! (» - l)!]4 det-4"=(-)- 2! (2,-)! "

3 a set of test matrices 55 In [ j it was shown that for the finite segment Bn of order n of Hilbert's matrix, so that [!2! (n - l)!]4 det2?"= IB!.;. (2«-l)! ' det^ = {-\)"-ln det Bn. The value of dn may be estimated quite easily using (2.2) and Stirling's formula n\ ~ (ra/e)n(27m)/2; the result is with <b{n) of the form Cow2, Co denoting a constant. Table la. Values of Table d(an) (520-02)- ( )- (3052-lO2)" ( )- ( )- (43-037)- ( )- lb. Measures of condition M(An) P(An) A listing of the values Sn for n =,2,, 0, rounded to 20 significant figures, is given in Table la. The extremely rapid increase of Sn is worthy of note; it serves to underline the poor "condition" of the An, as measured in terms of dn. As will be shown later other, more suitable measures of condition, M{A) = n- 4 \\A- \\A\\ = max Itti P(A) = \\max(a)/\min(a)\, are indicative of the same phenomenon. Next a few remarks about the inverses All the elements of An~x are

4 56 a set of test matrices integers. In fact, it can be shown that for i =,2,, we have =(-»-<(- while for i = 2, 3, and j =, 2,, n, we have where ('; ')('; v)-- Let us proceed to the characteristic roots Xt- and characteristic vectors Xi of 4, arranging the X,- according to increasing absolute value: Xi < X2 < < X ; we shall set Xi = Xm, Xn = X.v. Let the vector x< have the components x,-,-, j =, 2,, n. Since det {A - XI] = (-l^xex- - (Si a i)x"-2 + ]+ (-)»*.-\ it follows immediately that there is always at least one positive characteristic root. It seems that there is only one positive characteristic root, the others being negative. The existence of a positive characteristic root also follows from the result of Frobenius [2] that the root \m of largest absolute value of any positive nondecomposable matrix is positive and simple, and may be associated with a proper vector having positive components only, the only vector possible that possesses this property. Estimates for the X, are readily calculated from the following theorem due to Parker [3]: Let U = (iiij) be an arbitrary matrix of order n. Put n fi n = «_l \uu\ ; Pi = *<r v Qi = ; i=l J=l... J?*l «ft] Si = Pi + I uu, 5 = max; Si; Tj = Qj + [ [ ; T = max, 7/. FÄe«/or every characteristic root X of U (2.5) X - < min (5, F). Furthermore, the number way be replaced by any other number. Applying (2.5) to the matrix A, for = 0, it is seen that the maximum column sum T is a bound: (2.6) \mw < log (» + ) + % with y = denoting Euler's constant. Bounds for the components of xm may be obtained by means of Ostrowski's

5 A SET OF TEST MATRICES 57 theorem [4]: Let An = (0,7) be a matrix of order n of positive elements. Put n Ri = da, i =, 2,, n; R = max R{, r = min i?,: j i i i k\ = min au, k2 = min a,-,-; <ri = [(/ ki)/(r &i)]/2. Further, let the positive components xm, of the characteristic vector xm belonging to \m be normalized so that Then provided R 9^ r. In our case XMn S Xm, Ti- S ' ' ' S XM2 S XM = k2/ {R r + k2) < XMn < CTl n R = n, ki = (2«- l)-\ k2 = (2m - 2)", r = (m + j - l)". j-i It is found that (2.7) (2m2)-!C + O(m-0] < xmn < Dog 2/nJ^[\ + O(»-»)] The characteristic roots of least absolute value Xm are frequently also of interest. If the Xi(n) are distinct, then An' = (l/x/^xi«/-/»> i=l where the Xi(n) denote the characteristic column vectors of An belonging to X<(n), and the r/n) are the characteristic row vectors of the transposed matrix A T belonging to Xi(n). The smallest root Xm(n> may then be found as the dominant root /im(s) of An'- If it is further known that the roots X of the matrix U = {ui3) are real, then (2.5) permits the conclusion X > min (S, T); if, in addition, the particular root X = um is negative, then clearly (2.8) um- < [ - min (5, T) Another upper bound for Xm is provided by the fact that for definite matrices Thus (m 4 )-i < [M-'il < \\m\-\ (2.9) X. I < WA-^-K 3. The inverses and determinants. The first ten inversions were carried out by the method of partitioning [5]. A tabulation of the inverse 46_ computed by this method is given in Table 2.

6 58 a set of test matrices Ar = Table 2. The inverse of A$ Certain interesting properties of these inverses have already been pointed out before. The following features of the A _ are also noteworthy: a. It is seen that for j > I, j >, for n even sgn 4"} = \ (-l)w-i for i >, j > 2, for n odd ( {)*+> for i >, j =, for n odd. a +iia+ or c4"i+i with k + < «for» > 3. Now»I Ci24 Inspection of the [ 4 _ suggests that (3.) \\An-!\\ ~ C226" with C2~ With these estimates it is seen that the measures M(An) are approximately (3.2) M{An) ~ C2n26", so that the order of ill-conditioning is about that of Hilbert's matrix. The exact values of the M(An) are shown in Table lb. It has been stated [6] that "average" matrices have Af-condition numbers of the order n2 log n, and P-condition numbers of the order n. In the light of such orders for condition numbers the matrices A must indeed be called extremely pathological. There is a large number of techniques in matrix algebra which are specifically restricted to symmetric positive matrices H; for the use of such procedures there have been calculated the matrices From the reciprocals of 4, those of Hn are easily calculated as (3.3) i?,- = 4 -'u -r. The inverses HjT thus also consist of integer elements only. It is known (Taussky [8J) that the condition (in several senses) of A A T is worse than that of A. Obviously in Hk+ = (h$+)), (3.4) = + [(* + l)(j + I)]-* + [(* + 2)(j + 2)]-' + + l(i + k)u + k)tl - * }+ [(«+ *) O'**)!"'

7 a set of test matrices 59 for all t, j < k, k, 2,, with &4* = - Consequently, S<H-«= +*'(;) - (* + j)-2, i-k+i where ^(x) = T'(x + l)/r(x + ). Further, for fixed k, (3.5) max hf = hfl = + *'() - ( + j)" i,;' j * A listing of He is provided in Table 3. 7T2/6 - (* + j)-2. Table 3. 77*e matrix ij6 f Characteristic roots and vectors. Since there is a unique characteristic root Xji/n) of largest absolute value the power method [7] may be used to isolate XAf(rl>, while simultaneously obtaining the associated characteristic vector Xmm : (4.) AnX^ - lim Xitw \ (») (») A/fe Xk T lim xkm = xm(n), Table 4. Characteristic roots \mm \jn> "2 io-3 0"5 IO"6 0"8 0"9 io- io-2

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9 a set of test matrices 6 provided the initial arbitrary vector Xo(n) is not orthogonal to xm(n). The rate of convergence increases with the ratio between \m{n) and the next characteristic root of A. The absolutely largest elements of An occur in the first row; it is therefore advisable to normalize the xmm by keeping its first element equal to unity. In carrying out the computation a considerable saving in labor may be further achieved by utilizing the values of xm<n) obtained at the nth step to start the next step with a vector Xo("+) whose first n components were identical with Xmm. Proceeding in this manner there was obtained in seven or less iterations an agreement to six decimal places in both characteristic roots and vectors. The characteristic roots Xm'"' are shown in the middle column of Table 4; as indicated by (2.6) they actually do not grow faster than log n. The characteristic vectors Xmm belonging to the Xjf(n) are listed in Table 5a. Their smallest components Xm seem to decrease as n~l. The bounds given in (2.7) thus seem a bit wide. The characteristic roots Xm(n) = uu{n) of least absolute value were also calculated, together with their characteristic vectors xm(n), now normalized to have their last component equal to unity. The values of \m(n) are shown in the third column of Table 4. Here the inequality (2.8), with = 0, is found to restrict the roots fairly well, the upper bounds min (5(n), J^"')- being, at least for n < 0, less than 30 per cent too high. For purposes of comparison the resulting condition numbers P(An) were also computed; they are exhibited in the last column of Table lb. These suggest the following rough estimate: (4.2) P(An) ~ C326«log n, C3 m 8-0-', which, with (3.2), gives, (4.3) P(An) ~ (2/») log nm(an); Todd [] found for Hilbert's matrix B AVCO Manufacturing Stratford, Connecticut Corporation Trn~lM{Bn) < P(Bn) < *M(B«% Mark Lotkin. John Todd, "The condition of the finite segments of the Hilbert Matrix," in Contributions to the Solution of Systems of Linear Equations and the Determination of Eigenvalues, NBS Applied Math. Series No. 39, 954, p G. Frobenius, "Über Matrizen aus positiven Elementen," Sitzungsberichte Kgl. Preuss. Akad. Wiss., 908, p ; 909, p W. V. Parker, "Characteristic v. 5, 948, p roots and the field of values of a matrix," Duke Math. Jn., 4. A. Ostrowski, "Bounds for the greatest latent root of a positive matrix," London Math. Soc, Jn., v. 27, 952, p Mark Lotkin & Russell Remage, "Scaling and error analysis for matrix inversion partitioning," Ann. Math. Stat., 24, 953, p by 6. John Todd, "The condition of a certain matrix," Cambridge Phil. Soc., Proc, v. 46, 949 p R. A. Frazer, W. J. Duncan, & A. R. Collar, Elementary Matrices, Cambridge Univ. Press, Olga Taussky, "Note on the condition of matrices," MTAC, v. 4, 940, p. -2.