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Full text of "The QR-algorithm"

LIBRARY OF THE 

UNIVERSITY OF ILLINOIS 

AT URBANA-CHAMPAIGN 

5\0.8^- 

It 63c 
™>. II -2.0 




ENfiwuEamfi : 



AUG 51976 

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sponsible for its return to the library from 
which it was withdrawn on or before the 
Latest Date stamped below. 

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result in dismissal from the University. 

UNIVERSITY OF ILLINOIS LIBRARY^AJ LJRBA^AfHAMPAIGN 




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INFERENCE ROOM 



meed Computation 



ERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 

URBANA. ILLINOIS 61801 



CAC Document No. 12 
THE QR-ALGORITHM 

by 

Masako Ogura 
September 1, 1971 



Digitized by the Internet Archive 

in 2012 with funding from 

University of Illinois Urbana-Champaign 



http://archive.org/details/qralgorithmOOogur 



CAC Document No. 12 



THE QR- ALGORITHM 

by 

Masako Ogura 



Center for Advanced Computation 
University of Illinois at Urbana- Champaign 
Urbana, Illinois 6l801 



September 1, 1971 



This work was supported in part by the Advanced Research Projects 
Agency of the Department of Defense and was monitored by the U.S. 
Army Research Office - Durham under Contract No. DAHC04-72-C-0001. 



ABSTRACT 



The implementation of QR-algorithm on ILLIAC IV is described. An 
ASK subroutine for computing all eigenvalues of a real Hessenberg matrix of 
order less than or equal to 6k by this algorithm is attached. The QR-trans- 
formation consists of the decomposition of the matrix A^ into the product of 
a unitary matrix Q, and an upper triangular matrix R , and forming A, by 
post-multiplying R by Q, , where A. = A is the original matrix. All eigen- 
values are either isolated on the diagonal or are eigenvalues of a 2 x 2 
diagonal submatrix as k-> «> . 



TABLE OF CONTENTS 

Page 

1. INTRODUCTION 1 

2. USAGE 2 

2.1 Calling sequence 2 

2.2 Core storage used 3 

2.3 Constant 3 

3. EXAMPLE k 

k. QR-ALGORITHM 5 

k.l Brief outline of the QR-algorithm 5 

k.2 Practical computation 6 

5- PROGRAM DESCRIPTION 8 

5-1 Search for negligible subdiagonal elements 8 

5-2 Shifts of origin 9 

5-3 Search for two consecutive small subdiagonal elements ... 10 

5-h Double QR-transformation 11 

5-5 Computation of eigenvalues 12 

5.6 The number of iteration required 13 

REFERENCES lU 

APPENDIX 1 15 

APPENDIX 2 20 



INTRODUCTION 



An ASK program, for finding all eigenvalues of a real Hessenberg 
matrix of order less than or equal to 6k is written and tested on the B5500 
simulator. The QR- algorithm for real Hessenberg matrices described herein 
is that of Martin et al [2]. The ILLIAC IV computer time required for per- 
forming one iteration (computation of A from. A, , refer to k.l) on a 6k x 
6k matrix is approximately 10 millisecond. 

The necessary information for using this program, is given in 
Section 2. The test result of this program on a k x k real Hessenberg matrix 
is given in Section 3« In Section k, the outline of the QR-algorithm is 
given and Section 5 is devoted to the actual programming technique to imple- 
ment this algorithm on the ILLIAC IV computer. The flow chart and ASK program 
are attached as Appendices 1 and 2 respectively. 



2. USAGE 



This program assumes that the given real Hessenberg matrix is stored 
in the core memory in the straight storage scheme so that each row is stored 
across the PE's, starting with PEO. The real and imaginary parts of the eigen- 
values found are to be stored in the two rows in the PE memory specified by 
the user. The original matrix is destroyed and replaced by the matrix which 
results from the QR transformations. The content of ACAR2 and ACAR3 are 
destroyed since the ACAR3 is used for linkage between the subroutine and main 
program and ACAR2 is used for passing the address of parameters to the 
subroutine. 

2.1 Calling sequence 

Calling sequence for this subroutine is: 

CALL HOP (N, A, WR, WI, IT) 
where A designates the first row of the matrix and is declared in the main 
program as 

A: DATA a Q0 , a Q1 , , a^^, (0.0)M, 

a i0> a il ' ' a l,H-l, (0 '° )M ' 



Yi,o' Vi,i' ' • ' ' Vi,n-i' (0,0)M; 

where M = 6k - N. 
N is the size of the matrix declared as 

N: EQU 6k; 
or 

DEFINE N = 6k ##; 
or given as an integer, i.e., 6k. 

WR and WI are the rows in the PE memory where resulting real and imaginary 
parts of the eigenvalues respectively are to be stored. They are declared as 

WR: BLK 1; and WI: BLK 1; 
IT is the row vector in memory to which the number of iterations required for 
finding each eigenvalue is to be placed. This is declared as 

IT: BLK 1; 



3 
If zero is placed in place of IT, it is to be considered that a user does not 
want to know the number of iterations required. The CALL macro should be 
defined in the user's program as: 

DEFINE CALL &NAME (&PARAMETERS) = 
&IF &SIGN (&MFIELD(&NAME)) 

&THEN EXTERNAL &NAME; &FI 
&IF &EMPTY (&PARAMETERS) &TKEN &ELSE 
BEGIN BLOCK 

BEGIN USE (63) 
LIST: DATA & PARAMETERS 
END; 
CLC(2); 
SLIT (2) LIST; 
END; &FI 
CLC(3); 

SLIT (3) &NAME; 
EXCHL(3) &ICR ##; 

2.2 Core storage used 

This routine uses 500 words of PE memory for storing instructions. 
One row is used for storing PE numbers and three additional rows are used 
for temporary storage. ADBO ~ 32 are also used. 

2.3 Constant 

EPS (e), the constant which is used to test the convergence (k.l), 
is taken as 10" in this program. If a user wants to change the value of this 
constant, he may insert EPS: DATA (desirable value); in place of 

EPS: DATA @ - 10; . 



3. EXAMPLE 



matrix: 



A test of this program, was made on the B5500 simulator for the 



5-0 


-2.0 


-5.0 


-1.0 


1.0 


0.0 


-3-0 


2.0 


0.0 


2.0 


2.0 


-3-0 


0.0 


0.0 


1.0 


-2.0 



with € = 10 . The comparison of the eigenvalues obtained by this program to 
the exact values is given in Table 1. The selection of this small matrix and 
a relatively large e was made because of the speed of the SSK simulator on 
the B5500; the execution speed ratio of the simulator to the ILLIAC IV is 
approximately 1:10 . 

Table 1 



Eignevalues obtained 


Exact eigenvalues 


on B5500 simulator 




3-999999997867 


k.O 


I.OOOOOOOOIO66 + 1. 99999999992 8i 


1.0 + 2.0i 


I.OOOOOOOOIO66 - 1. 99999999992 8i 


1.0 - 2.0i 


-1.000000000000 


-1.0 



h. QR- ALGORITHM 

^.1 Brief outline of the QR-algorithm 

The QR-transformation consists of the decomposition of the matrix 
A into the product of a unitary matrix Q and an upper triangular matrix R , 
and forming A by post-multiplying R by &. Thus 

\ + i - W where \ ■ \\' (1) 

therefore 

where A = A is the original matrix. It can be shown in general that A 

(k) (k) 

tends to a form in which a. ' . a. ' . n = for i = 0, 1, . . . . , N - 3 

as k increases. All eigenvalues are therefore either isolated on the diagonal 
or they are eigenvalues of a 2 x 2 diagonal submatrix. The amount of calcula- 
tions involved in a QR step is greatly reduced if the matrix A is in the 
Hessenberg (or almost triangular) form. Since there are several stable 
methods available to reduce a general matrix to this form (ASK program. HSBG 
is written for this purpose), the QR-algorithm. is used after such reduction. 

In order to achieve rapid convergence, it is essential that the 
origin shifts be applied and that each shift be close to an eigenvalue 
of the matrix. The QR-algorithm with shift of an origin s is expressed 
as: 

\ + 1 - \ \ + \ z where \ - V = W < 2 > 

or in other words 

\ + i ■ \ \V 

However, even when A is real, some of the eigenvalues may be 

complex. If the transformation (2) is carried out with a complex value of 

s , A is in general a complex matrix. This deficiency can be overcome 

by performing two steps of (2) with shifts of s and s respectively. 

Since s. and s, . are both real or complex conjugate in this transformation, 
k k + 1 7 

A should be always real. This transformation is described as 



and 



<\\ ♦ 1> < R k + 1 V = <\ " S?> <\ - s k + 1 1 ) (3) 



One method of calculating A^ by (3) would be to form the real 
matrix r = (A - si) (A, - s i), computing its unitary- triangular 
decompositon to obtain Q, Q and transform A, by means of this, thus 
giving A^ p . This process requires a prohibitive amount of work, but it is 
shown [l] that when the matrix is in the Hessenberg form, it is unnecessary 
to compute more than the first column of r, and that this immediately gives 
the transformation to be applied to A^. 

k.2 Practical computation 



If (3) is rewritten as 

\ + 2 = w * A k w and W*r = A 

where W = Q, Q. . and A is the triangular matrix R, R , W is a unitary 
matrix which reduced r to the triangular A, and W is composed of N unitary 

Fi 1 ° " 

factors of the form M. = so that W = M, M p M. . From the 

L i-" 

form of each M. we see that the first column of W is equal to the first 

column of IVL , and this is any unitary matrix, the transpose of which 
eliminates the elements of the first column of V below diagonal. 

Since we wish to transform A^ to A by W, we first operate on 
A with M . This will change the first three rows and columns of A^ since 
the first column of r contains only three non-zero elements. It follows; 



V 



00 



"10 



01 



11 



21 



l 02 * 



'12 



22 ' 



. . a 



0,N - 1 

1,N - 1 

l 2,N - 1 



'N- 1, N -1 



-» 



00 



01 



02 



a io 


!ii 


a !2 


a 20 


a 21 


tk 


a 30 


a 31 


a 32 



"03 



°13 



£3 
"33 
l 43 



0,N 
l l,N 
l 2,N 



3,N - 



N - 1, N - 1 



where the elements changed by the row and column operations are underlined 
and primed respectively. The resulting matrix is no longer in the Hessenberg 



form and since 



\ 



+ 2 



is in the Hessenberg form, we can say that the matrices 



M 2 . M 3 



. . NL reduce M. A^ M to the Hessenberg form. In the practical 
computation, therefore, it is necessary to compute only the first column of r. 

After each iteration (we call the calculation of A, _ from A 

v k + 2 ± 

an iteration), all subdiagonal elements of A, p are examined to see if any 
of them are "negligibly small." If so, the eigenproblem. for the current 
matrix splits into that for two or more Hessenberg matrices of smaller sizes, 
and the iterations continue with the submatrix in the bottom right-hand 
corner. It may happen that while no individual subdiagonal element is 
sufficiently small to be regarded as negligible, the product of two consecu- 
tive elements may be small enough to permit us to work with a submatrix. 
Therefore the examination of the matrix A is performed to see if any 
two consecutive subdiagonal elements are small. 



5- PROGRAM DESCRIPTION 

5.1 Search for negligible subdiagonal elements 

We assume that the size of the matrix under consideration is 
(n + l) x (n + l) where n takes integer values between 1 and N - 1. If the 
last negligible subdiagonal element is in position (£, £ - l), it is required 
only to work on the submatrix in the rows and columns £ to n. If none of the 
subdiagonal elements are negligible, I is taken to be 0. The following 
criterion is used 

|a J, J-il < e ^ a i - l, i- J + lag, jl); 
This criterian examines whether a, £ & _ is negligible compared to the local 
diagonal elements. 

On each PE £, the following computations are simultaneously 
performed: 

f(i) S-|a if £ _ ± \ - e (\, £ _ ^ £ _ 1 | + | a ^ £ \ 

for i = 1, 2, . . . , n. 

If f(i) is negative, 1 is placed in ith bit of the ACAR. Then 
searching is made for the lowest bit of the ACAR which contains 1. For 
example, in the following case: 

1 2 3 ... 19 20 63 

ACAR 



1 0...0 1 0...0 



I is set to be 20. 



Then test is made if £ = n or Z = n - 1. If £ = n, one eigenvalue 
is found in the place (n, n) and the matrix is deflated by 1, and n is 
decreased by one. In the case that £ = n - 1, two eigenvalues are found as 
the eigenvalues of the bottom-right hand corner 2x2 submatrix. Then two 
columns and rows are deflated and n is decreased by 2. 



r > 



5.2 Shifts of origin 

The shifts of origin at each stage are taken to be the two roots, 

s. and s, n , of the 2x2 matrix in the bottom right-hand corner of the 
kk+1 ° 



current iL. This gives 



s. + s. t = a 

k k + 1 n 



1, n - 1 



+ a 



n, n 



and 



s, s. _ = a 

kk+1 n 



1 _, a - a n a 

1, n - 1 n,n n - 1, n n, n-1 



(h) 



In some rare cases, the process fails to converge with these shifts 
of origin. An example of such failure is provided by matrices of the type: 















1 " 


1 
















1 















1 




1 







Here the shifts of origin, given in (k) , are both zero, and since the matrix 
is orthogonal, it is invariant with respect to the QR transformation without 
shifts. However, if one iteration is performed with any shifts of origin 
which are loosely related to the norm, of the matrix, the convergence is very 
rapid. Therefore in the case where ten iterations do not produce an eigen- 
value, the usual shifts s. and s, _ are replaced by shifts defined by 
' k k + 1 



s. + s. ., = 1.5 ( a n + a _ 
k k+1 v ' n, n-1 1 'n-1, n 

2 



2 



) • 



s, s. 



= (U. 



I) 



(5) 



k k + 1 Vl n, n - l 1 ' n - 1, n - 2 1 
This strategy is used again after 20 unsuccessful iterations. If 30 
unsuccessful iterations are needed then a failure indication is given. 

In this program, ITS is the name of an ADB where the iteration 

count is stored. When ITS +- 10, 20, we form S = s. + s. , and Y = s. s, . 

' kk+1 kk+1 

on a PE n according to (k) and store in ADB's. When ITS = 10 or 20, scheme 
(5) is used for computing S and Y. If ITS = 30, it is assumed that this 
algorithm fails to produce eigenvalues. As a result, only eigenvalues 
computed prior to this point are given in WR and WI. 



10 

5-3 Search for two consecutive small sub diagonal elements 

After determining &, (5-l), the submatrix in the rows 1 to n are 
examined to see if any two consecutive subdiagonal elements are small enough 
to work with an even smaller submatrix. To test if we are to start at the 
row m, we compute the elements p , q and r such that 

p=a -a {s. + s, _ ; + s. s, , + a a 
m mm. mm k k + r k k + 1 m, m + 1 m + 1, m 

q. = a (a + a _ , - s, - s, n ) (6) 

m m + 1, m v mm m + 1, m + 1 k k + l y K ' 

m" m + 2 , m + 1 m + 1, m. " 
The criterion applied is 

i I C f <3 I + l r I ) 

- 1 tti ' m ' 



m, m 



(7) 



< € |p | (|a | + la ! + la 

— Ir m' m + 1, m + l 1 ' m,m' ' m - 1, m -l| ) 

where we test whether or not the elements which appear in the positions 

(m. + 1, id), (m + 2, m + l) are negligible compared with the three local 

diagonal elements a _ _ , a and a 

m + 1, m + 1' m,m m - 1, m - 1 

Here we take m to be the largest integer (> i) for which condition (6) is 

satisfied. 

For this computation, the mode bits are turned on for PE I through 
PE n. The p m , q^ and r for i = 1, I - 1, . . . , n are computed according 
to (6) on all PE's whose mode bits are turned on, and comparison is made to see 
whether (7) is satisfied. If (7) is satisfied on PE m, 1 is placed in the 
mth bit of the ACAR. Then the search is made for the lowest bit of the 
ACAR which contains 1, and m is set equal to this bit number. If no 1 is 
found in the ACAR, m. is taken to be i. 



11 

5.+ Double QR-transformation 

A is computed by applying the QR-double transformation to iL 

in such a way that A. =N . . . . N A, N . ... IT 

k + 2 n m k m n 

* 

* U i U i 

where N. = I ?r and U. = (p. + t . , q. , r . , . . . .0). 

l „ 2 l XJr i — i ; ^i' i' 

1 



Here 



and 



p. = a. . - a. . (s. + s. . ) + s, s, _ + a. . . a. 
i n ii k k + 1 k k + 1 1,1 +1 i + 1,: 

q. = a. _ . (a. . + a. _ . . - s, - s, n ) 
*i i + l,i v ii i+l,i+l k k + r 

r. = a. _ . _ a. _ . for i = m 

l i + 2,i + l i + l,i 



p. = a. . + t. , q. = a. , . 

i i,i-l—i' l i + 1, l - 1 

and r. = a. _ . _ for i * m. 

l l + 2,i - 1 T 



t. and 2K. are defined as 
i i 



/2 2 2 

t. = + ./p. + q. + r. 
x — v J. 1 1 

2 2 — 

2KT = t. + p.t. . 
i ill 

Row modification: 

* (i) 
For i = m, m + 1, . . . , n, the elements of N. A . = 
' ' ' ' i k 

N* (N. , • . . N* A, N . . . N. ,) are different from those of A\ X ' in 
ii-l m k m l-l k 

only three rows, i.e., ith, (i + l)th and (i + 2)th rows. These new elements 
are computed in the following way with the elements of A denoted by a, . : 

(i, j ) - element = a. . - [(p. +t.)a.. +q.a. . . +r.a. _ .]— 



(i+l.j ) - element =a. n . - [(p. +t.)a..+q.a. n . + r.a. _ .] ^ 

V '° 1+1,0 1 ~ 1 1J 1 1 +1 ,J 1 1+2, J pir 2 



;i+2,j) - element = a. + ^ . - [(p. + t.Ja.. + q.a. + ^ . + r.a. + ^ . } _fi 

2K. 

l 

for j = i,i + 1, ...,n 



12 

2 
In the actual computation, t. and 2K. are first computed and then 

p , q. and r. are found and stored in ADB's. The mode bits of PE's which 
*V 1 l 

contain a. .,a. . .,, . . . . ,a. are turned on and 
1,1 i,i+l i,n 

c. : (p. + t. )a. . + q.a. _ . + r.a. _ . are computed on these PE's. The 
l - v i - i 13 i i+l, 3 i i+2,J 

computation of new elements 

c. 

(i.j) - element = a. . - t— 
v ' ij t. 

e.g. 

-i -*"i 

(i.j+l) - element = a. . ,. - ?r 

1 > J + 1 2K 
i 

c.r. 
(i,j+2) - element = a. ± . 2 ^ 

2K . 

l 

are then performed. 

Column Modification: 

/ * (Ox * (i) 

Similarly (N. A v )N. is computed from N.A v ' for l = m, m + 1, 

* (k) 
. . . , n in the following way where the element of matrix N.A. are 

denoted as a . , : 
3,h 

(d,i) - element = a.. - [ (p. + t.)a.. + Vj,i+1 + r i a j,i+2 ] \ 

q i 

(j.i+l) - element = a. . , - [(p. + t.)a.. + q.a. + r.a. ] 5 

Vd ' ' 3,i+l v *i - i' ji *i 3,i+l i 3,i+2 2K 2 

i 



(3,1+2) - element = a - [ (p. + t.)a.. + q^^ + r^^] -ig 



for j = i, . . . • , min [i+3,n] . 

As in the row modification, c! = (p. + t. )a. . + q.a . . + r.a are 

l i — l 31 1 3,1+1 1 3,1+^ 

computed on the PE's which contain a., for 3 from i through min [i + 3, n], 

then the computations of new (3*,i) - element, (3, i + l) - element and 

(j, i + 2) - element are performed on the corresponding PE's. 

5 • 5 Computation of eigenvalues 

The eigenvalues are calculated as the last step of program after 

a. , . or a. . , a. , . become negligibly small for all 0<i<2i-2. At each 
1+1,1 i,i-l 1+1,1 to 



13 



time when the matrix is deflated by 2, 1 is placed in position n of ADB named 
SOLV. If the transformation is carried out successfully, SOLV looks like the 
following, 

0123^56789 10 62 63 



SOLV 



001000010010 1 



Here, the eigenvalues we are looking for are a^_, a__, a, , , a,.,., 
t & & qq> 22> 4V 55' 



a 88> a ll,ll' 



, a,-, £, and eigenvalues of the following 2x2 matrices 



11 



"22 



%6 



77 



'99 



10, 10 



and 



a 



'62,62 



l 63,63 



These eigenvalues can be calculated simultaneously on the corresponding PE's. 



5.6 The number of iteration required 

After each deflation, the number of iterations required to find each 
eigenvalue (or two eigenvalues) appears in ADB named ITS. This is placed as 
the nth element of the row vector specified by the user as the fifth para- 
meter in the calling sequence. If this parameter is specified as 0, the 
number of iterations is not stored anywhere. 



REFERENCES 



1. J.G.F. Frances: "The QR Transformation — A Unitary Analogue to the LR 
Transformation/' Parts I and II, Camp. Journal , h_, 265-271 and 332- 
3>+5 (1961/62). 

2. R.S. Martin, G. Peters and J.H. Wilkinson: The QR Algorithm for Real 
Hessehberg Matrices. Numer. Math ., lj+, 219-231 (1970). 

3. B.N. Parlett: The LU and QR Algorithm. Mathematical Methods for 
Digital Computers, Vol. 2, A. Ralston and H. Wilf, Editors, J. Wiley, 
1968. 

k. J.H. Wilkinson: The Algebraic Eigenvalue Problem. Oxford University 
Press (1965)- 



APPENDIX 1 



15 



n «- size of matrix - 1 
clear rows IT and SOLV 




Yes 




ITS *- 



look for single 
small subdiagonal 
element and find I 




Yes 



Yes 



Yes 



FIN, 




ONEW] 




TWOWJ 




A I] 



Yes 



No 



computer origin shifts 

V s k + i and 

s = s k + s k + l 
Y = s k s k + l 



compute origin shifts 

S = L ' 5 ( l a n,n-ll + I a n-l,n- 2 ') 

Y;: Cla^ x | +|a n .^ n . 2 D 2 



ITS <- ITS + 1 



16 



i «- i + 1 



no 



1 



look for two consecutive 
small sub diagonal 
elements and find m 



i <- m 




yes 



no 



p, q, r were already 
computed in 

FINDM 



compute p, q and r 



compute t and 2K 



_v- 



row modification 
compute new element 
a . . , a. n ., a. _ 

io' i+i, a' 1+2, j 

for j = 1, . . . , i 



^ 



column modification 
compute new element 

a. .. a. . « a. . ~ 

D,i D,! 4 "! a, 1+2 

for j = i, . • • , n 




IT 




(oimT) 

-^ fifth argument 

calling 



no 



store ITS into row IT 
as an nth element 



±k- 



n <- n - 1 <; 



( NEXTW J 



yes 



-* 




yes 



no 



store ITS into row IT 
as an nth element 



store 1 into nth bit 

of SOLV 



ik_ 



n <-n - 2 



C NEXTWJ 



19 




FIN 



mode bit <- 1 for all PE's 
■which satisfy 

< PEN < N - 1 



find eigenvalues on 
all enabled PE's 



WR <~ real part of 
eigenvalues 

WI «- imaginary part 
of eigenvalues 




FAIL 



mode bit «- 1 for all PE's 
which satisfy 

n < PEN < N - 1 



end of this 
subroutine 



APPENDIX 2 



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UNCLASSIFIED 



Security Classification 



DOCUMENT CONTROL DATA R&D 

(Security elaaallleatlon ot till*, body of ahatraci and Indmmhtg annotation mutt ha antarad whan tho ovarall report la ctmaalllad) 


1. originating activity f Corporate autfior) 

Center for Advanced Computation 
University of Illinois at Urban a- Champaign 
Urbana, Illinois 61801 


aa. REPORT SECURI TV CLASSIFICATION 

UNCLASSIFIED 


2b. GROUP 


3. REPORT TITLE 

THE QR-ALGORITHM 


4. DESCRIPTIVE notii (Typa of report and rnelualra dataa) 

Research Report 


5- authorisi (Flrat turn, middle Initial, laat noma) 

Masako Ogura 


« REPORT DATE 

September 1, 1971 


7a. TOTAL NO. OF PACES 

kk 


7b. NO. OF REFS 


•a. CONTRACT OR GRANT NO. 

USAF 30-(602)-4l¥r 

b. PROJECT NO. 

ARPA Order 788 

c. 
d. 


•a. ORIGINATOR*! REPORT NUMBER'S) 

CAC Document No. 12 


•b. OTHER REPORT xoisi (Any othar numbara that may ba aaalgnad 
thla raport) 


10 DISTRIBUTION STATEMENT 

Copies may be requested from the address given in (l) above. 


II. SUPPLEMENTARY NOTES 

None 


12. SPONSORING MILITARY ACTIVITY 

Rome Air Development Center 
Griffiss Air Force Base 
Rome, New York I3UU0 


13. ABSTRACT 

The implementation of QR- algorithm on ILLIAC IV is described. An 
ASK subroutine for computing all eigenvalues of a real Hessenberg matrix of 
order less than or equal to 6k by this algorithm is attached. The QR- 
t ran s format ion consists of the decomposition of the matrix A into the product 
of a unitary matrix Q, and an upper triangular matrix R, and forming A by 
post-multiplying R by Q , where 1 = A is the original matrix. All eigen- 
values are either isolated on the diagonal or are eigenvalues of a 2 x 2 
diagonal submatrix as k -*■ °° . 



DD ,'•<," .1473 



UNCLASSIFIED 

Security Classification 



UNCLASSIFIED 



Security Classification 



KEY WO KOI 



Matrix Algebra 



ROL E W T 



UNCLASSIFIED 



Security Classification