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L161 — O-1096
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 aQ0, aQ1 , , a^^, (0.0)M,
ai0> ail ' ' al,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 = 0 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> <Rk + 1 V = <\ " S?> <\ - sk + 11) (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*Akw 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, Mp 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
l02 *
'12
22 '
. . a
0,N - 1
1,N - 1
l2,N - 1
'N- 1, N -1
-»
00
01
02
aio
!ii
a!2
a20
a21
tk
a30
a31
a32
"03
°13
£3
"33
l43
0,N
ll,N
l2,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
M2. M3
. . 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
|aJ, J-il < e^ai - 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-|aif £_±\ - 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:
0 1 2 3 ... 19 20 63
ACAR
0 0 1 0...0 1 0 0...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:
0
0
0
0
1 "
1
0
0
0
0
1
0
0
0
1
0
1
0
0
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-11 'n-1, n
2
2
) •
s, s.
= (U.
I)
(5)
k k + 1 Vl n, n - l1 ' n - 1, n - 21
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 + ly K '
m" m + 2 , m + 1 m + 1, m. "
The criterion applied is
i I C f <3 I + lr I )
- 1 tti ' m '
m, m
(7)
< € |p | (|a | + la ! + la
— Irm' m + 1, m + l1 ' 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 pm, 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. 0=N . . . . N A, N . ... IT
k + 2 n m k m n
*
* UiUi
where N. = I ?r and U. = (p. + t . , q. , r . , 0 . . . .0).
l 0„ 2 l XJri — 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 + riaj,i+2] \
qi
(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 0 1
Here, the eigenvalues we are looking for are a^_, a__, a, , , a,.,.,
t & & qq> 22> 4V 55'
a88> all,ll'
, a,-, £, and eigenvalues of the following 2x2 matrices
11
"22
%6
77
'99
10, 10
and
a
'62,62
l63,63
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 *- 0
look for single
small subdiagonal
element and find I
Yes
Yes
Yes
FIN,
ONEW]
TWOWJ
A I]
Yes
No
computer origin shifts
V sk+i and
s = sk + sk+l
Y = sk sk+l
compute origin shifts
S = L'5 (l an,n-ll + I an-l,n-2')
Y;: Cla^ x| +|an.^n.2D2
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
0 < 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