A Simple Set of Test Matrices for Eigenvalue Programs*

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1 Simple Set f Test Matrices fr Eigenvalue Prgrams* By C. W. Gear** bstract. Sets f simple matrices f rder N are given, tgether with all f their eigenvalues and right eigenvectrs, and simple rules fr generating their inverses in the nnsingular cases. In general, these matrices are nnsymmetric. They can have sets f duble and triple rts. In each f these cases, tw f the rts f the dublet r triplet can crrespnd t a single eigenvectr. The general frm f the N X N matrix is : _ ± ± - where the ± is in the Jth clumn f the first rw and in the (iv + K)th clumn f the last rw. The first rw can be expressed as () et ± e, where e< is the vectr cnsisting f a unit in the ith psitin. The special case in which the first rw is simply et will be shwn t be equivalent t taking J = with a negative sign in (). (Ntatinally this des nt make sense, but it will save writing t include it in this way.) The last rw can be similarly written as etn-i ± e^+i-k- Nte that J and K and the signs may be different. The eigenvalues and eigenvectrs f these matrices may all be expressed in the frm : Eigenvalue = cs a. Right eigenvectr = [sin (c + a), sin (c + a), sin (c + Na)]T. The values f a and c are given belw. In this discussin, all rman letters represent integers, sets, r matrices. efine the sets f a and c values by Tables I and II. The eigenvalues and eigenvectrs are then the unin f the sets f a and c values specified in Table III. This unin gives all TV eigenvalues and as many eigenvectrs as exist. Received pril 5, 968, revised June 7, 968. * Wrk supprted in part by the U. S. tmic Energy Cmmissin under grant EC T( ) 469 and by the rgnne Natinal Labratry. ** epartment f Cmputer Science, University f Illinis, Urbana, Illinis. 9

2 C. W. GER Table I. Sets f a Values Set Name a Members l prr/in + - J - K) lúp<n+l-ik + J)/ (p - )t/(7v + -J-K)lúpúN+l- BI prr/j Ú p < J/ ik + J)/ 5 (p - )t/j lúp^j/ Cl V-k/K è p < K/ C (p - l)ir/k up è K/ I) E Table II. Sets f c Values Set Name X YI c Members (it - Jet)/I -Ja/ (at - (N + - K)a)/ These first fur sets are generated by crrespnding sets i" -(V + - K)a/ f a values. r/ Prfs. The prf f each statement is trivial but tedius; therefre nly the duble psitive case will be explred in detail. The ith element,- f the eigenvectr is sin (c + ict). The t'th element f vl is: sin (c 4- Oi - l)a) + sin (c + Oi + l)a), == cs a sin iœ + ia), =,- cs a. The first element f is if < i < TV, sin (c + a) ± sin (c 4- Ja). We wuld like this t be equal t cs a. Nte that s that we want sin c 4- sin (c 4 a) t,- cs a () sin a = ±sin (c 4 Ja). Similarly, we want

3 SIMPLE SET OF TEST MTRICES FOR EIGENVLUE PROGRMS (3) sin (c + in + l)a) = ±sin (c + ( V + - K)a). () and (3) frm a pair f equatins which determine a and c such that cs a and are an eigenvalue and eigenvectr f. Table III. Eigenvalues Sign f extra ne digit in rw rw N a Values 4 4 l BI Cl Eii K,J bth even + BI C EU K dd, / even 4 B Cl Eii K even, J dd B C E if K, J bth dd Crrespnding c Values YI I Y X YI X X Y X Nte Nte Nte 3 Nte Nte Nte 3 Nte Nte Nte 3 Nte Nte Nte 3 Nte : If and B r and C sets have a nnnull intersectin, each a in the intersectin crrespnds t a duble rt X with a single eigenvectr. slutin f i )p = is given by: p. = ( l/sina) [cs (c + a),cs (c + a), -, N cs (c + Na)]T. Nte : If the B and C sets have a nnnull intersectin, each a in the intersectin crrespnds t a duble rt with tw eigenvalues. The secnd can be fund by adding x/ t the c given in the table. Nte 3: If J and K d nt satisfy the cnditin, set E is nt included. Nte that if the first rw is simply et, then sin c = replaces Eq. (). This is identical t requiring that () hld with a minus sign and J =. We will prceed by slving () and (3) and shwing that all N eigenvalues have been accunted fr. Cnsider nly the case with plus signs in () and (3). Equatin () implies that either (4.) CO + p7r = c + Ja, r (4.) c = (p + IV - (c + Ja), while Eq. (3) implies that either (5.) c + in + l)a + qir = c + (TV + - K)a, License r r cpyright restrictins may apply t redistributin; see

4 C. W. GER (5.) c 4 (X + l)a = (gx + ) - (c + (T 4- - K)a). One f each f these pairs must be used. (4.) and (5.) imply that: (6) a = pir/j = qir/k, c arbitrary. These cases will usually be included in ther cases, but tw special cases shuld be nted nw. If p = q =, a =. c can be chsen as 7r/. This gives set which nly ccurs in the ++ case. The eigenvectr is [,,, -, l]r. If J, K are bth even, p = «7/, q = K/ gives a = -n. gain, c can be chsen as 7r/. This gives a set E. The eigenvectr is [,,,, -]T. (4.) and (5.) imply that (7) c = (p + )tt - Ja= (<74 l)x - (V + - K)a. Let g = ; since p is arbitrary, then (8) a = pir/in + - J - K). p = has already been handled, p = ÍN + J K)/ has been handled (equality case) r is equivalent t smaller p. (a shuld remain in the range [, it].) Therefre ^ p < (X + J K)/ and this case gives a set l. Frm (7) we get c set since multiples f it may be discarded in c. (4.) and (5.) imply that (9) a = pir/j, () c = (g + )tt - (V + - K)a. gain we can take j p < J/ and q = t get a set BI and c set YI. Similarly (4.) and (5.) lead t a set Cl and c set. We nw have a set f values and vectrs. The remaining prblem is t shw that there are n thers. First nte that we have N values in Table III. In the + + case, fr example, we have sets,b,c, and E if J and X are even. This gives a ttal f [(V + -K-J- l)/] + [(J - l)/] + UK - l)/] + (+ if J,K bth even). Fr each f the fur cases we have J K Number f Members dd dd N + - ÍK + J)/ - + (J - l)/ 4 (K - l)/ + = N dd even N + - (X 4 J)/ - i + (J - l)/ + (X - )/ + = X even dd X + - (X + J)/ - \ + (J - )/ + (X - l)/ + = X even even X 4 - (X + J)/ - + (J - )/ + (X - )/ 4 + = X Therefre, if the a sets d nt intersect, we have all f the values and vectrs. If, fr example, sets l and BI have a nnempty intersectin, we have:

5 SIMPLE SET OF TEST MTRICES FOR EIGENVLUE PROGRMS 3 (I with a = p!7r/(x + - J - K) = ptr/j, lúpi<n+l-ij + K)/, lt%p<j/. This implies that J and (X + J K) have a cmmn factr f at least 3. In this case cnsider where If < i < N, v = ( cs al)n pa = i cs (c + ia)/ sin a vi sin a = Oi ) cs (c 4 Oi l)a) i cs a cs (c 4 ia) + ii + ) cs (c + (z + l)a), = [cs (c(i l)a) + cs (< + ii + l)a) cs a cs (c + ia)] + cs (c + (i + l)a) cs (c 4 Of l)a), = sin a sin (c 4 ia), = sin a,-. Therefre, Fr i =, we have v% = <. i<i sin a = cs (c 4- a) cs a cs (c + a) + J cs (c 4 Ja). Frm (), Ja = pir, and in this case the c value can be btained frm the set. Therefre, cs (c 4 Ja) = COS Hw Ja)/ + Ja) = cs ((it 4 Ja)/) = cs (tt + p7r)/ =, which implies that vi = i. Similarly, we can shw that vn = at. Thus, if sets and 5 intersect, we get a duble rt X with a single vectr and a slutin f ( \I)p = as given abve. similar result hlds fr intersectins f l and Cl. If 5 and Cl intersect, then a = p,7r/j = pir/k. This crrespnds t Eq. (6), which means that c is arbitrary. By taking the c given in Table II and 7r/ plus that c, tw linearly independent eigenvectrs are btained. If l, 5, and Cl have a cmmn intersectin, then a triple rt with tw eigenvectrs and ne ther principal vectr will be btained. Therefre, we have btained a cmplete set f eigenvalues. Example. N = 8, X = J = 6, duble psitive case. Matrix J

6 4 C. W. GER Set l Set 5 Set Cl Set) SetX a Values tt/3 x/3 tt/3 x/3 tt/3 tt/3 k/ -ât/ c Values -3x/ -3tt/ T lt w/ k/ Eigenvalues,,, - The inverses f a set f related matrices,, are easy t cmpute. (These matrices can ccur in ne space dimensin bundary value prblems. Their eigenvalues are ( cs a).) The 7th clumn f the inverse has the frm a b if < I < N [w + a, w -p a,, and, w + la, w + Ia + b,, w + Ia + (X - 7)6]7 [w + a, w + a,, w + Na]T if I = r X, where the w, a and b must be chsen (separately fr each I) t satisfy equatins arising frm the first and last rws f the matrix, (w, a and 6 may be nninteger). Thus, fr X = 6, =, J = 5 in the \- case, the matrix and its inverse are and - L respectively. These results can be extended t cver the types f matrices arising frm differential equatins in tw r mre space dimensins. We will treat the extensin frm ne t tw dimensins. Cnsider the blck matrix 5 = n ± Where the matrix is as defined earlier, the matrix is a diagnal matrix with elements 8 and where the extra ±5 blcks appear in the J'th and X' 4 Xth blck clumns, r d nt appear at all. Cnsider the vectr ±) () V = [mi, P%, -, MT' ], where is an eigenvectr f crrespnding t X. We will write this as p. X. The tth blck f Bv fr < i < N' is. pi-ii, + m 4 5/ii+i = iißi-i + ßi+i)/pi + X)/ui. Hence, if the first and last blcks can be fixed apprpriately, and if

7 SIMPLE SET OF TEST MTRICES FOR EIGENVLUE PROGRMS 5 (3) 5(m.-i 4 ßi+i)/ßi + X = X', Ki<N', v will be an eigenvectr f 5 crrespnding t X'. (3) is a recurrence relatin fr pn with a slutin (4) ßi = sin (c' 4 ia), when X' = cs a' + X. The first blck f 5i- leads t the requirement that (5) M = ±ßj' while the last blck requires that (6) M-v'+i = ±pn'+i-k' (5) and (6) determine c' and a' by use f Tables I, II, and III, s that the eigenvalues and eigenvectrs f 5 can be determined frm (). In the case f repeated rts withut a cmplete set f eigenvectrs, the principal vectrs can als be used fr p. r. They will give rise t principal vectrs f 5. If p. and are the single eigenvectrs assciated with duble rts a' and a respectively, and if p and are the assciated principal vectrs f rder, then it is trivial t shw that p. X, p X are principal vectrs f rders and 3, while p X m X and p. X are independent eigenvectrs. The extensin t any number f dimensins is straightfrward. The use f mre dimensins allws multiple rts t be intrduced. Fr example, the 5X5 matrix = l_. has rts,,, and. If the prcess abve is applied twice, we get a matrix f rder 5 with at least 7 zer rts and principal vectrs f rder 4. The authr wuld like t acknwledge the helpful cmments f Prfessr Fsdick and the referee. epartment f Cmputer Science University f Illinis Urbana, Illinis 68

Cambridge Assessment International Education Cambridge Ordinary Level. Published

Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid