NOLTA, IEICE
Paper On the generation of very ill-conditioned integer matrices
Tetsuo Nishi 1 a), Siegfried M. Rump 2 ,1 b), and Shin’ichi Oishi 1 ,3 c)
1 School of Science and Engineering, Waseda University 3-14-9 Okubo, Shinjuku, Tokyo, 169-0072 Japan
2 Institute for Reliable Computing Hamburg University of Technology, 21071 Hamburg, Germany
3 CREST, JST
a) [email protected] b) [email protected] c) [email protected]
Abstract: In this paper we study the generation of an ill-conditioned integer matrix A =[aij] with |aij|≤μ for some given constant μ.Letn be the order of A. Wefirstgivesomeupper bounds of the condition number of A in terms of n and μ. We next propose new methods to generate extremely ill-conditioned integer matrices. These methods are superior to the well-known method by Rump in some respects, namely, the former has a simple algorithm to generate a larger variety of ill-conditioned matrices. In particular we propose a method to generate ill-conditioned matrices with a choice of desirable singular value distributions as benchmark matrices. Key Words: ill-conditioned matrix, condition number, verified numerical computation, integer matrix, singular value distribution, companion matrix
1. Introduction Extremely ill-conditioned matrices are required to examine the quality of verified numerical compu- tations for solving simultaneous linear equations [1–6]. Here an ill-conditioned matrix means that its condition number Cond(A)=AA−1 is 1016 or larger in IEEE 754 binary64 (double precision) arithmetic. Though the condition number of a matrix is the most important index in numerical anal- ysis, its properties such as upper bounds for integer matrices discussed in this paper have not been fully investigated [1, 12–14]. Once S. Rump [7] proposed a method to generate extremely ill-conditioned matrices with floating- point entries. The method is now most well-known and a modification is used as the standard tool to generate an ill-conditioned matrix in INTLAB (see “randmat ”). However, since his method is based on Pell’s equation (which is well-known in number theory) with a limited number of solutions, there is not so much variety of matrices to be generated. Therefore we are seeking alternative methods to obtain a greater variety of ill-conditioned matrices. Moreover we are interested in apriorispecifiable singular value distributions. Up to now several methods [8–11] which may be considered as extensions of Rump’s method [7] were proposed. In this paper we give new generation methods of ill-conditioned 226 [Received: May 27, 2010. Revised: November 30, 2010. Published: April 1, 2011.] [DOI: 10.1588/nolta.2.226] Nonlinear Theory and Its Applications, IEICE, vol. 2, no. 2, pp. 226N245 ©IEICE 2011 integer matrices1. Section 2 introduces to the subject and summarizes related previous results. Throughout the paper μ denotes a large positive integer such as 108,1016 or 253 (but μ = 10 or less may also, theoretically, be permissible). We are aiming to generate extremely ill-conditioned n × n integer matrices A =[aij] with |aij|≤μ. In Section 3 several upper bounds of the 2-norm condition number are shown in terms of n and μ, in particular for specific distributions of the singular values of A. The results in Section 3 are closely relatedtoSection5.InSection4wegiveanewgenerationmethod2 for extremely ill-conditioned matrices. The method may be regarded as a modification of Rump’s method [7] and has the following features in comparison with Rump’s original method [7] and its extensions [8–10]: (i) the algorithm is simpler, (ii) the obtainable condition number is roughly the same as for previous methods, and (iii) it can generate a greater variety of matrices. The obtained matrices are somewhat similar to a companion matrix. Some numerical examples are shown for illustration. The matrices generated up to Section 4 in this paper as well as those by previous methods [7–11], in particular by Rump’s method, bear the drawback that the first n − 1 singular values are large and not far from the first one, the spectral norm, whereas only the n-th singular value is extremely small. This implies that the inverse of those matrices is very near to a matrix with rank one (see examples in Section 4). This may not necessarily be preferable for a benchmark matrix. Therefore we consider in Section 5 some desirable distributions of singular values of a matrix and give partial solutions to the generation problem by using some special types of matrices. We will see that the condition number of the obtained matrices is nearly optimal in some sense. Numerical examples show good agreement with theory.
2. Preliminaries The main purpose of this paper is to generate an n × n integer matrix A =[aij] which satisfies
|aij|≤μ (i, j =1, 2, ··· ,n)(1) and whose condition number Cond(A)=AA−1 is extremely large3. It is obvious that the maximally achievable condition number of A increases monotonically with μ. In this paper we assume4
|det(A)| =1. (2)
Note that Eq. (2) can be achieved only under very delicate relations among the entries aij,in particular when n and μ are large. We will briefly summarize previous results.
2.1 Rump’s method[7] One of the key points of Rump’s method is to find a 2 × 2 integer matrix V s.t. PkQ PkQ V , V , = QP det ( )=det QP =1 (3) where k is a prescribed small positive integer5.TheintegersP and Q are very large (depending on the desired condition number), such as 1050, and are chosen to satisfy Pell’s equation
1Though our final matrices are to be floating-point matrices, we discuss about the generation of integer matrices due to the same reason as in [7]. 2The matrices generated in this paper as well as in the previous ones [7–9] are very sparse. For example, there are few elements equal to ±1, and the number of nonzero elements not equal to ±1isonlyaboutcn,wherec =2orc =3.For practical use we can derive dense matrices by multiplying them by appropriate matrices P with |det(P )| = 1 from left and/or right. We can also add rows or columns by similar operations. Note, however, that this may change the condition number. 3 n The infinity norm A ∞ defined by maxi j=1 |aij | is often used in verified numerical analysis, but the spectral norm A 2 is also used in this paper. Similarly, both Cond∞(A) and Cond2(A) are used. See also Eqs. (10) and (11). 4This is not a strict condition, but proved to be important to generate ill-conditioned integer matrices. 5The symbols n and V in Eqs. (1) and (3) are different from those in the Rump’s paper [7]. The order “2(n +1)”of thematrixin[7]iswrittenas“n” in this paper. 227 P 2 − kQ2 =1, (4) from which det(V ) = 1 in Eq. (3) follows. Utilizing V in Eq. (3), Rump constructed an n × n (n is even) integer matrix A in Eq. (1.6) of [7] and showed by rather tricky calculations that
−1 2 Cond∞(A)=A∞A ∞ ≥ (P + kQ) . (5)
Under the reasonable assumption that P , Q ∼ μn/2 we have
2 n Cond∞(A) ≥ (P + kQ) ∼ 4μ . (6)
This shows that Cond2(A) can be extremely large in Rump’s method.
2.2 Previous extensions of Rump’s method Rump’s algorithm can be generalized by replacing V in Eq. (3) with the following two kinds of matrices [8–10].
2.2.1 Replacing V by more general type of a 2 × 2matrix The matrix V in Eq. (3) was generalized into PF V , V PG− QF . = QG det( )= =1 (7)
For prescribed large integers P and Q having no common factor, e.g., P =2k,Q=3m , (8) P =2k1 5k2 11k3 ,Q=3m1 7m2 we can find F and G satisfying Eq. (7) by using the extended Euclidean algorithm. The equations corresponding to Eqs. (5) and (6) also hold for this case. This method greatly enlarges the class of generated matrices.
2.2.2 Replacing V by a 3 × 3matrix The discussion similar to Eqs. (3) and (7) is possible for this case.
3. Upper bounds of condition number of matrices in terms of µ
In this section we show some upper bounds of Cond2(A), the condition number of a matrix A with respect to the spectral norm6,intermsofμ. As before, the general assumptions for this section are
| det(A)| =1 and |aij|≤μ for 1 ≤ i, j ≤ n. (9)
As we see, the condition numbers with infinity norm and the spectral norm are related by [12, 13]: √ √1 A ≤A ≤ nA n 2 ∞ 2 (10) ∴ 1 A ≤ A ≤ n A . nCond2( ) Cond∞( ) Cond2( ) (11)
This means that concerning large condition numbers both norms give almost the same value. So in this paper we use both norms conveniently; we use mainly the spectral norm in Sections 3 and 5 and the infinity norm in Section 4. Though our aim is to generate ill-conditioned integer matrices, we discuss on real matrices in most of Sections 3 and 5. 6 A Ax The spectral norm is the matrix norm induced by the Euclidean vector norm, i.e. 2 := maxx=1 . It follows T A 2 = (A A), where denotes the spectral radius. 228 A µ 3.1 Condition number of in terms of for special singular√ value distributions T Let the eigenvalues of AA be λi > 0(i =1, 2, ··· ,n), so that σi ≡ λi (i =1, 2, ··· ,n)arethe singular values of A. We use decreasing order, i.e.
λ1 ≥ λ2 ≥ λ3 ≥···≥λn−1 ≥ λn(> 0), (σ1 ≥ σ2 ≥ σ3 ≥···≥σn−1 ≥ σn > 0) (12)
Here λi are the solutions of the characteristic equation
det(λ1 − AAT )=λn − tr(AAT )λn−1 + ···+(−1)ndet(AAT )=0, (13) where “tr(X)” denotes the trace of a matrix X and “1” denotes the identity matrix. Note that T 2 tr(AA ) is equal to the Frobenius norm AF .Using|aij|≤μ it follows n n 2 T 2 2 2 λi = AF =tr(AA )= aij ≤ n μ , (14) i=1 i,j=1 and therefore A2 = σ1 = λ1 ≤AF ≤ nμ. (15)
An obvious estimation for Cond2(A)uses σ σ n−1 σ σ n−1 σ n −1 A 1 1 i=1 i 1 i=1 i σ σ ≤ σn ≤ nnμn. Cond2( )=σ = n σ = | A | = 1 i 1 (16) n i=1 i det( ) i=1
This can be improved as follows, in particular allowing to use information not only on the maximum size of the elements |aij|, but also on their individual size and distribution. For any positive definite n×n-matrix B, Hadamard’s determinant inequality [Horn/Johnson: Matrix Analysis, Theorem 7.8.1] yields n det(B) ≤ bii. i=1 −1 For any matrix B with | det(B)| =1wehave|(B )ii| = | det(B(i))|,whereB(i) is the matrix B after deleting the i-th row and column. Applying this to the positive definite matrix B := AAT and using that B(i) is, as a principal submatrix, positive definite as well, shows T −1 T 2 n−1 ((AA ) )ii ≤ (AA )jj ≤ (nμ ) , (17) j=i so that −1 2 T −1 T −1 n 2(n−1) A 2 = (AA ) 2 ≤ tr((AA ) ) ≤ n μ . (18) Putting things together yields the estimation
−1 n/2+1 n Cond2(A)=A2A 2 ≤ n μ . (19)
An even better estimation is obtained by Theorem 1(Guggenheimer, et al. [14], 1995): For an arbitrary real n × n matrix A we have