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Extending quark propagators in time
Volume 184, number 1
PHYSICS LETTERSB
22 January 1987
E X T E N D I N G Q U A R K P R O P A G A T O R S IN T I M E C.B. C H A L M E R S , R.D. K E N W A Y and D. R O W E T H Physics Department, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
Received 9 October 1986 The distant source method (DSM) computes quark propagators on large lattices via two or more calculations on smaller lattices. Here it is tested in quenched QCD. Systematic errors are shown to be acceptably small. However, the D S M is not competitive with other algorithm s for generating quark propagators from scratch; its usefulness lies in permitting the extension of an existing set of propagators in time.
The distant source method (DSM) [1] relies on using several quark propagator calculations on small lattices to calculate the quark propagator, with Dirichlet boundary conditions in time, on larger lattices. This is potentially useful because it alleviates some of the constraints imposed by limited computer m e m o r y and speed. This is because less data is needed per computation, and also because numerical algorithms tend to converge faster for smaller systems. The method is not directly competitive in terms of the total amount of computer time used, with the solution of the full system of equations on the large lattice by means of an efficient algorithm such as iterative block successive over relaxation (IBSOR) [2], when both are started from scratch. However, DSM is useful when a set of propagators (with Dirichlet boundary conditions in time) already exist and analysis shows that it is desirable to extend the lattice size in the time direction. Use of the D S M does involve a systematic error, which is due mainly to finite size effects on the small lattice. It has already been shown that this is insignificant for free fermions [1]. Here we will demonstrate that the systematic error introduced by the D S M in quenched Q C D is much smaller than the statistical error in an average over four gauge configurations. This systematic error grows with decreasing quark mass, but is likely to remain acceptably small, relative to the level of statistical accuracy of present hadron mass calculations, whenever finite size affects are themselves acceptably small. The equation for the quark propagator x in temporal gauge on a lattice of N timeslices with Dirichlet boundary conditions in time is [2]
D+Mx=6,
(1)
which may be partitioned [2] as
(D 1 + M) -T
T (D 2 + M) -T
Xl
T (D 3 + M)
T --T
(DN i + M)
T
-T
(D N + M )
0370-2693/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
X2
~2
X3
~3
XN- 1
~N
(2) 1
, XN
63
PHYSICS LETTERS B
Volume 184, nUmber 1
22 January1987
Here D t is the three-dimensional lattice covariant derivative for Susskind fermions on timeslice t, M = ml and T.,. = 1 ( _ 1).1+.2+.38..,. We assume that the source is on one of the first J timeslices. With
DI+M
T
0
...
0
-T 0
0
All =
A12 =
0
0 T
0
T 0
...
0
P,t+M
-T
' P J + I -}- M
o
T;o
A21 -;
T
-T 0 (3)
A22 = ..•
0
0
0 and x z = (x I . . . . . xs), yX = (Xj+ 1. . . . . AllX + A 1 2 Y = ~,
XN),
...
0
-T
T ~N --F M
we then have
A21X + A22Y = 0.
(4)
Assuming we know X l , . - - , x j f r o m a previous calculation, we m a y solve for y since
Txs ]
and this is similar to the system of eq. (2) but with Txj used as the source• We could continue to divide the p r o p a g a t o r equation into more than two sets of blocks, in an obvious way. In practice it is necessary to perform the first calculation on a larger lattice than j4, because the estimate for xj c o m p u t e d on a j 4 lattice would be contaminated b y (time) b o u n d a r y effects, and so would be a bad choice for the source in the next computation. Thus we take x j to be well away f r o m the time b o u n d a r y on the first lattice• For example, with N = 24 and J = 8, we calculate x a on a 164 lattice, and use Tx 8 as the new source for a calculation on timeslices 9-24• A schematic picture of this is shown in fig. 1. It was shown in ref. [1] that this m e t h o d accurately reproduces the analytic results in the free fermion case, at mass of 0.2. Tables 1 - 4 show the results for quenched Q C D at a r a n g e of quark mass values. The D S M results are c o m p a r e d with those obtained using the iterative block S O R algorithm [2]. This analysis was done using four 164 gauge configurations at B = 6.0, each separated b y 1792 pseudo-heatbath sweeps [3]• The distant source method was used to solve for the quark p r o p a g a t o r on a 163 × 24 lattice (in which the gauge fields were periodically extended in time) in two stages. The original 8-function source was placed On timeslice 5. The two resulting sets of linear equations for 164 lattices were solved using the conjugate gradient algorithm, but any appropriate algorithm could have been used. At the second stage of 64
Volume 184, number 1
PHYSICS LETTERS B first on
calculation
I to 16
I
I
ii
o
ii
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22 January i987
,
2
,
4
,
6
,
8
1
,0
,
,
,0
14
1
,
18
,
20
2
,
,o, I
,
24
II
oII I second on
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9 to 2 4 ; T x 8 is t h e source
Fig. 1. Distant source method for obtaining 163 × 24 propagators by performing successive calCulations on 164 lattices.
t h e D S M c a l c u l a t i o n , i n w h i c h a 16 4 p r o p a g a t o r is e x t e n d e d t o a 16 3 x 24 p r o p a g a t o r t h e C G a l g o r i t h m w a s f o u n d t o c o n v e r g e m o r e s l o w l y t h a n f o r t h e o r i g i n a l 16 4 c a l c u l a t i o n . T h e i t e r a t i v e l y c o m p u t e d r e s i d u a l d e p a r t e d s i g n i f i c a n t l y f r o m t h e a c t u a l r e s i d u a l at all m a s s v a l u e s , p r o b a b l y d u e t o r o u n d i n g e r r o r s , a n d it p r o v e d n e c e s s a r y t o c o u n t e r t h i s b y r e s t a r t i n g t h e C G c a l c u l a t i o n , as s h o w n i n t a b l e 5, u s i n g t h e l a t e s t a p p r o x i m a t i o n f o r t h e q u a r k p r o p a g a t o r as t h e f i r s t g u e s s . In tables 1 and 2 we present the results from one gauge configuration for the pseudoscalar and vector m e s o n p r o p a g a t o r s a t a m a s s o f 0.09. T h e t w o sets o f d a t a f o r t h e D S M a n d t h e I B S O R a r e i n v e r y g o o d a g r e e m e n t . A t h i g h e r q u a r k masses the results are of the s a m e or b e t t e r quality; at lower q u a r k m a s s e s t h e y
Table 1 Results from one configuration, comparing the pseudoscalar (PS) meson propagator using the DSM and the iterative block SOR algorithm on a 163 x24 lattice, and the conjugate gradient on a 164 lattice. The quark mass is 0.09 and fl = 6.0 Time
DSM PS
IBSOR PS
C G 16 4 PS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.8288E- 01 0.1014E + 00 0.3033E + 00 0.8493E + 00 0.2724E + 01 0.8355E + 00 0.2994E + 00 0.1115E+00 0.4570E - 01 0.1939E- 01 0.8733E- 02 0.3987E- 02 0.1894E- 02 0.9092E- 03 0.4493E - 03 0.2155E- 03 0.1030E- 03 0.5022E - 04 0.2414E - 04 0.1176E- 04 0.5239E-05 0.2975E - 05 0.1101E- 05 0.1120E- 05
0.8291E- 01 0.1014E + 00 0.3033E + 00 0.8493E + 00 0.2725E 4- 01 0.8355E 4- 00 0.2995E + 00 0.1115E+00 0.4576E - 01 0.1951E- 01 0.8595E- 02 0.3971E- 02 0.1884E- 02 0.9039E - 03 0.4463E - 03 0.2140E- 03 0.1022E- 03 0.4981E - 04 0.2394E- 04 0.1166E- 04 0.5193E 05 0.2948E - 05 0.1091E- 05 0.1109E- 05
0.8288E- 01 0.1014E + 00 0.3033E + 00 0.8493E + 00 0.2724E + 01 0.8355E + 00 0.2994E + 00 0.1115E+00 0.4570E - 01 0.1957E- 01 0.8534E- 02 0.4049E- 02 0.1815E- 02 011009E - 02 0.3777E - 03 0.3779E- 03
'65
Volume 184, number 1
PHYSICS LETTERS B
22 January1987
Table 2 A s table 1 b u t f o r the v e c t o r (VT) m e s o n p r o p a g a t o r . Time
DSM VT
IBSOR VT
C G 164 V T
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
0 . 2 9 0 7 E - 01 0 . 4 8 9 2 E - 01 0 . 1 3 8 6 E + 00 0 . 6 6 8 3 E -- 00 0 . 2 1 1 4 E + 01 0 . 8 7 9 1 E + 00 0.1389E+00 0 . 7 2 8 3 E - 01 0 . 1 6 3 5 E - 01 0 . 7 1 2 0 E - 02 0 . 2 6 2 2 E - 02 0 . 1 6 1 8 E - 02 0 . 4 5 3 4 E - 03 0 . 2 5 6 2 E - 03 0 . 7 4 6 2 E - 04 0 . 2 6 3 0 E - 04 0.1136E- 04 0 . 5 6 7 4 E - 05 0 . 2 0 4 0 E - 05 0 . 1 5 6 0 E - 05 0 . 4 3 5 1 E - 06 0 . 3 6 3 2 E - 06
0 . 2 9 0 9 E - 01 0 . 4 8 8 7 E - 01 0 . 1 3 8 6 E + 00 0 . 6 6 8 3 E + 00 0 . 2 1 1 4 E + 01 0 . 8 7 9 0 E + 00 0.1389E+00 0 . 7 2 7 7 E - 01 0 . 1 6 3 9 E - 01 0 . 7 3 6 8 E - 02 0 . 2 6 0 9 E - 02 0 . 1 6 1 3 E - 02 0 . 4 5 1 4 E - 03 0 . 2 5 5 0 E - 03 0.7421E - 04 0.2612E - 04 0.1129E- 04 0 . 5 6 3 4 E - 05 0 . 2 0 2 7 E - 05 0 . 1 5 4 8 E - 05 0 . 4 3 2 6 E - 06 0 . 3 6 0 5 E - 06
0.2907E0.4892E = 0.1386E + 0.6683E + 0.2114E + 0.8791E + 0.1389E+ 0.7283E0.1635E0.7410E 0.2566E 0.1655E 0.4205E0.2889E0.5760E 0.6478E -
01 01 00 00 01 00 00 01 01 02 02 02 03 03 04 04
Table 3 T h e a v e r a g e d v e c t o r (VT) m e s o n p r o p a g a t o r m e a s u r e d u s i n g f o u r c o n f i g u r a t i o n s o n a 163 X 24 lattice at fl = 6.0 using, in c o l u m n 1 the D S M , in c o l u m n 2 the iterative B l o c k S O R a l g o r i t h m a n d in c o l u m n 3 the c o n j u g a t e g r a d i e n t a l g o r i t h m o n a 164 lattice. T h e q u a r k m a s s is 0.09. Time
DSM vector
IBSOR vector
C G 164 v e c t o r
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0 . 1 8 6 6 E - 01 0 . 4 1 7 5 E - 01 0 . 1 1 8 9 E + 00 0 . 8 7 6 4 E + 00 0 . 1 7 8 8 E + 01 0.8116E+00 0 . 1 0 7 5 E + 00 0 . 5 0 6 7 E - 01 0 . 1 0 6 5 E - 01 0 . 4 0 8 7 E - 02 0 . 1 4 4 5 E - 02 0 . 7 4 5 1 E - 03 0 . 2 3 9 1 E - 03 0.1135E-03 0 . 3 6 9 4 E - 04 0 . 1 3 9 9 E - 04 0 . 5 8 4 7 E - 05 0 . 2 7 7 8 E - 05 0 . 1 0 0 9 E - 05 0 . 6 5 9 5 E - 06 0 . 1 2 3 0 E - 06 0 . 1 5 4 8 E - 06 0 . 3 5 9 6 E - 07 0 . 5 4 7 5 E - 07
0 . 1 8 6 7 E - 01 0 . 4 1 7 3 E - 01 0.1189E + 00 0.8764E + 00 0 . 1 7 8 8 E + 01 0.8115E+00 0.1075E + 00 0 . 5 0 6 4 E - 01 0 . 1 0 6 6 E - 01 0 . 4 1 9 8 E - 02 0 . 1 4 3 5 E - 02 0 . 7 4 3 1 E - 03 0 . 2 3 8 2 E - 03 0.1130E-03 0.3672E- 04 0.1390E- 04 0 . 5 8 1 2 E - 05 0 . 2 7 5 9 E - 05 0 . 1 0 0 1 E - 05 0 . 6 5 3 9 E - 06 0 . 1 2 1 9 E - 06 0 . 1 5 3 6 E - 06 0 . 3 5 6 5 E - 07 0 . 5 4 3 3 E - 07
0.1866E0.4175E0.1189E+ 0.8764E + 0.1788E + 0.8116E+ 0.1075E + 0.5067E 0.1065E0.4222E 0.1416E0.7603E0.2254E0.1315E0.2669E0.3004E -
66
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Volume 184, number
1
PHYSICS
LETTERS
22
B
January1987
Table 4 A s table 3 but for a quark mass of 0.04 and including in column 4 the standard deviation in the IBSOR data. Time
D S M vector
IBSOR vector
C G 16 4
IBSOR
SDEV
1
0 . 2 2 8 5 D - 01
0 . 2 2 9 1 D - 01
0 . 2 2 8 5 D - 01
0.1271D - 01
2
0 . 4 2 8 5 D - 01
0 . 4 2 8 2 D - 01
0 . 4 2 8 5 D - 01
0.8747D - 02
3
0.1208D + 00
0.1209D + 00
0.1208D + 00
0 . 2 5 6 6 D - 01
4
0.8937D + 00
0.8940D + 00
0.8937D + 00
0.1980D +
5
0 . 1 7 0 8 D + 01
0 . 1 7 0 8 D + 01
0 . 1 7 0 8 D + 01
0.2822D +
6
0.8625D + 00
0.8619D + 00
0.8625D + 00
0.1226D + 00
7
0.1114D + 00
0.1116D + 00
0.1114D + 00
0.2681D - 01
8
0 . 6 3 2 7 D - 01
0 . 6 2 8 7 D - 01
0 . 6 3 2 7 D - 01
0 . 1 9 8 9 D - 01
9
0 . 1 4 5 3 D - 01
0 . 1 4 6 7 D - 01
0 . 1 4 5 3 D - 01
0.5356D - 02
10
0.5745D - 02
0.6915D - 02
0.7179D - 02
0.4284D - 02
11
0.3010D-
02
0.2584D - 02
0.2427D - 02
0.1711D - 02
12
0.2014D - 02
0.1957D - 02
0.2109D - 02
0.1836D - 02
13
0.7189D - 03
0.6937D - 03
0.5986D - 03
0.7021D - 03
14
0.3570D - 03
0.3446D - 03
0.4782D - 03
0.4497D - 03
15
0.1700D - 03
0 . 1 6 1 9 D - 03
0 . 1 0 7 8 D - 03
0.1913D - 03
16
0.4017D - 04
0.3825D - 04
0.7265D - 04
0.9151D - 04
17
0.1043D - 04
0.1022D - 04
0.8322D - 04
18
0.1300D - 04
0.1204D - 04
0.4104D - 04
19
- 0.3278D - 05
- 0 . 3 7 8 3 D - 05
0.2782D - 04
20
0.1233D - 04
0.1123D - 04
0.1695D - 04
21
- 0.8109D-
0.7149D-
00 00
07
- 0.2133D - 06
22
0.6788D - 05
0.6160D - 05
0.7709D - 05
05
23
0.1689D - 05
0.1566D - 05
0.1745D - 05
24
0.4467D - 05
0.4081D - 05
0.4173D - 05
are worse. Table 3 shows the averaged vector propagator obtained by the two methods at m = 0.09 (from the same four configurations) and as we might expect, the two sets of data are very close. We have chosen to display the vector meson propagator because it is sensitive to cancellations in the statistical average, and hence is more representative than the pseudoscalar propagator. What is apparent from table 4 is that agreement is not quite so good at lower values of the quark mass, although, as can be seen from the column of standard deviations, the discrepancy between the two methods on almost all timeslices is at least an order of magnitude smaller than the statistical error. A probable explanation for the disagreement is that the second "source" T x 8 becomes increasingly contaminated by finite time effects as the quark mass is lowered, due to the small lattice only having 16 timeslices. At a given quark mass, the discrepancy between D S M and the IBSOR results is most marked near the "join" in the D S M i.e. on timeslices 10 and 11 (as can be seen clearly in table 4). Notice that, because we
Table 5 Numbers of iterations used for a two-stage D S M calculation of Mass
a 163 × 24
quark propagator at various quark masses
m.
Number of iterations on first (164 ) stage
on second stage
0.50
40
60+
30
0.16
120
100+
50
0.09
150
150+
50
0.04
300
250 +
50
0.01
500 + 200
600 + 300
67
Volume 184, number 1
PHYSICS LETTERSB
22 January 1987
used the e v e n / o d d partitioned conjugate gradient algorithm [2], this effect sets in on timeslice 10 rather than on timeslice 9 which is the actual time boundary on the second, because timeslice 9 has the opposite " p a r i t y " to timeslices 8 and 10. The effect of this join becomes unobservable after two timeslices in our data. However, in high statistics measurements and at low quark masses, it m a y be sufficiently pronounced to require corrective action. We propose to replace the bad D S M timeslices (10 and 11) by the corresponding timeslices obtained for the first 164 lattice. In this latter data, the effect of timeslice 16 is relatively insignificant whenever finite-size effects are small, as is evident from table 4. To summarise then, we can see that the D S M works well as long as we do not lower the quark mass too far or, in general, provided finite-size effects are small. At the lower quark masses it might seem advisable to use three steps on a 164 lattice go that, say, x 6 could be used as the source in the second stage, and x12 in the third stage, to reduce finite-time effects. However, the method would probably then no longer be cost-effective: The results presented here demonstrate the accuracy of the D S M sufficiently clearly that we see no reason to extend the comparison with IBSOR to a larger statistical sample. The statistical error in state-of-the-art hadron propagator calculation is, of course, smaller than that in table 4. However, our results indicate that the systematic error in the D S M is only likely to become comparable with this in regimes where finite-size effects are already overwhelming. So far, the IBSOR has proved the more efficient way of obtaining 163 × 24 propagators on the D A P from scratch, and because these lattices appear to be long enough for our present analysis [4] (at least to the extent that signal-noise ratio problems begin around timesfice 19) there is no reason to extend these propagators. However, in future work with higher statistics it might be beneficial to extend the propagators in time to 163× 32 for example, permitting the inclusion of more timeslices in the fits to the asymptotic decay. Given the existence of a large set of 163 × 24 propagators, the D S M would be a more efficient way of accomplishing this than starting again from scratch. This work was done using the DAPs at Edinburgh which are supported by SERC grant N G 15 908. C.B.C. is supported by a Vans Dunlop Scholarshi p f r o m the University of Edinburgh and D.R. is supported by the SERC and Inmos Ltd.
References [1] R.D. Kenway, Phys. Lett B 158 (1985) 327. [2] C.B. Chalmers, R.D. Kenway and D. Roweth, Algorithms for calculating quark propagators on large lattices, Edinburgh preprint 86/361 (revised version), submitted to J. Comput. Phys. [3] N. Cabibbo and E. Marinari, Phys. Lett B 119 (1982) 387. [4] K.C. Bowler, C.B. Chalmers, R.D. Kenway, G.S. Pawley and D. Roweth, Hadron mass calculations using Susskind fermions at fl = 5.7 and 6.0, Edinburgh preprint 86/369, Nucl. Phys. B., to be published.
68
PHYSICS LETTERSB
22 January 1987
E X T E N D I N G Q U A R K P R O P A G A T O R S IN T I M E C.B. C H A L M E R S , R.D. K E N W A Y and D. R O W E T H Physics Department, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK
Received 9 October 1986 The distant source method (DSM) computes quark propagators on large lattices via two or more calculations on smaller lattices. Here it is tested in quenched QCD. Systematic errors are shown to be acceptably small. However, the D S M is not competitive with other algorithm s for generating quark propagators from scratch; its usefulness lies in permitting the extension of an existing set of propagators in time.
The distant source method (DSM) [1] relies on using several quark propagator calculations on small lattices to calculate the quark propagator, with Dirichlet boundary conditions in time, on larger lattices. This is potentially useful because it alleviates some of the constraints imposed by limited computer m e m o r y and speed. This is because less data is needed per computation, and also because numerical algorithms tend to converge faster for smaller systems. The method is not directly competitive in terms of the total amount of computer time used, with the solution of the full system of equations on the large lattice by means of an efficient algorithm such as iterative block successive over relaxation (IBSOR) [2], when both are started from scratch. However, DSM is useful when a set of propagators (with Dirichlet boundary conditions in time) already exist and analysis shows that it is desirable to extend the lattice size in the time direction. Use of the D S M does involve a systematic error, which is due mainly to finite size effects on the small lattice. It has already been shown that this is insignificant for free fermions [1]. Here we will demonstrate that the systematic error introduced by the D S M in quenched Q C D is much smaller than the statistical error in an average over four gauge configurations. This systematic error grows with decreasing quark mass, but is likely to remain acceptably small, relative to the level of statistical accuracy of present hadron mass calculations, whenever finite size affects are themselves acceptably small. The equation for the quark propagator x in temporal gauge on a lattice of N timeslices with Dirichlet boundary conditions in time is [2]
D+Mx=6,
(1)
which may be partitioned [2] as
(D 1 + M) -T
T (D 2 + M) -T
Xl
T (D 3 + M)
T --T
(DN i + M)
T
-T
(D N + M )
0370-2693/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
X2
~2
X3
~3
XN- 1
~N
(2) 1
, XN
63
PHYSICS LETTERS B
Volume 184, nUmber 1
22 January1987
Here D t is the three-dimensional lattice covariant derivative for Susskind fermions on timeslice t, M = ml and T.,. = 1 ( _ 1).1+.2+.38..,. We assume that the source is on one of the first J timeslices. With
DI+M
T
0
...
0
-T 0
0
All =
A12 =
0
0 T
0
T 0
...
0
P,t+M
-T
' P J + I -}- M
o
T;o
A21 -;
T
-T 0 (3)
A22 = ..•
0
0
0 and x z = (x I . . . . . xs), yX = (Xj+ 1. . . . . AllX + A 1 2 Y = ~,
XN),
...
0
-T
T ~N --F M
we then have
A21X + A22Y = 0.
(4)
Assuming we know X l , . - - , x j f r o m a previous calculation, we m a y solve for y since
Txs ]
and this is similar to the system of eq. (2) but with Txj used as the source• We could continue to divide the p r o p a g a t o r equation into more than two sets of blocks, in an obvious way. In practice it is necessary to perform the first calculation on a larger lattice than j4, because the estimate for xj c o m p u t e d on a j 4 lattice would be contaminated b y (time) b o u n d a r y effects, and so would be a bad choice for the source in the next computation. Thus we take x j to be well away f r o m the time b o u n d a r y on the first lattice• For example, with N = 24 and J = 8, we calculate x a on a 164 lattice, and use Tx 8 as the new source for a calculation on timeslices 9-24• A schematic picture of this is shown in fig. 1. It was shown in ref. [1] that this m e t h o d accurately reproduces the analytic results in the free fermion case, at mass of 0.2. Tables 1 - 4 show the results for quenched Q C D at a r a n g e of quark mass values. The D S M results are c o m p a r e d with those obtained using the iterative block S O R algorithm [2]. This analysis was done using four 164 gauge configurations at B = 6.0, each separated b y 1792 pseudo-heatbath sweeps [3]• The distant source method was used to solve for the quark p r o p a g a t o r on a 163 × 24 lattice (in which the gauge fields were periodically extended in time) in two stages. The original 8-function source was placed On timeslice 5. The two resulting sets of linear equations for 164 lattices were solved using the conjugate gradient algorithm, but any appropriate algorithm could have been used. At the second stage of 64
Volume 184, number 1
PHYSICS LETTERS B first on
calculation
I to 16
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ii
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22 January i987
,
2
,
4
,
6
,
8
1
,0
,
,
,0
14
1
,
18
,
20
2
,
,o, I
,
24
II
oII I second on
calculation
9 to 2 4 ; T x 8 is t h e source
Fig. 1. Distant source method for obtaining 163 × 24 propagators by performing successive calCulations on 164 lattices.
t h e D S M c a l c u l a t i o n , i n w h i c h a 16 4 p r o p a g a t o r is e x t e n d e d t o a 16 3 x 24 p r o p a g a t o r t h e C G a l g o r i t h m w a s f o u n d t o c o n v e r g e m o r e s l o w l y t h a n f o r t h e o r i g i n a l 16 4 c a l c u l a t i o n . T h e i t e r a t i v e l y c o m p u t e d r e s i d u a l d e p a r t e d s i g n i f i c a n t l y f r o m t h e a c t u a l r e s i d u a l at all m a s s v a l u e s , p r o b a b l y d u e t o r o u n d i n g e r r o r s , a n d it p r o v e d n e c e s s a r y t o c o u n t e r t h i s b y r e s t a r t i n g t h e C G c a l c u l a t i o n , as s h o w n i n t a b l e 5, u s i n g t h e l a t e s t a p p r o x i m a t i o n f o r t h e q u a r k p r o p a g a t o r as t h e f i r s t g u e s s . In tables 1 and 2 we present the results from one gauge configuration for the pseudoscalar and vector m e s o n p r o p a g a t o r s a t a m a s s o f 0.09. T h e t w o sets o f d a t a f o r t h e D S M a n d t h e I B S O R a r e i n v e r y g o o d a g r e e m e n t . A t h i g h e r q u a r k masses the results are of the s a m e or b e t t e r quality; at lower q u a r k m a s s e s t h e y
Table 1 Results from one configuration, comparing the pseudoscalar (PS) meson propagator using the DSM and the iterative block SOR algorithm on a 163 x24 lattice, and the conjugate gradient on a 164 lattice. The quark mass is 0.09 and fl = 6.0 Time
DSM PS
IBSOR PS
C G 16 4 PS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.8288E- 01 0.1014E + 00 0.3033E + 00 0.8493E + 00 0.2724E + 01 0.8355E + 00 0.2994E + 00 0.1115E+00 0.4570E - 01 0.1939E- 01 0.8733E- 02 0.3987E- 02 0.1894E- 02 0.9092E- 03 0.4493E - 03 0.2155E- 03 0.1030E- 03 0.5022E - 04 0.2414E - 04 0.1176E- 04 0.5239E-05 0.2975E - 05 0.1101E- 05 0.1120E- 05
0.8291E- 01 0.1014E + 00 0.3033E + 00 0.8493E + 00 0.2725E 4- 01 0.8355E 4- 00 0.2995E + 00 0.1115E+00 0.4576E - 01 0.1951E- 01 0.8595E- 02 0.3971E- 02 0.1884E- 02 0.9039E - 03 0.4463E - 03 0.2140E- 03 0.1022E- 03 0.4981E - 04 0.2394E- 04 0.1166E- 04 0.5193E 05 0.2948E - 05 0.1091E- 05 0.1109E- 05
0.8288E- 01 0.1014E + 00 0.3033E + 00 0.8493E + 00 0.2724E + 01 0.8355E + 00 0.2994E + 00 0.1115E+00 0.4570E - 01 0.1957E- 01 0.8534E- 02 0.4049E- 02 0.1815E- 02 011009E - 02 0.3777E - 03 0.3779E- 03
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Table 2 A s table 1 b u t f o r the v e c t o r (VT) m e s o n p r o p a g a t o r . Time
DSM VT
IBSOR VT
C G 164 V T
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
0 . 2 9 0 7 E - 01 0 . 4 8 9 2 E - 01 0 . 1 3 8 6 E + 00 0 . 6 6 8 3 E -- 00 0 . 2 1 1 4 E + 01 0 . 8 7 9 1 E + 00 0.1389E+00 0 . 7 2 8 3 E - 01 0 . 1 6 3 5 E - 01 0 . 7 1 2 0 E - 02 0 . 2 6 2 2 E - 02 0 . 1 6 1 8 E - 02 0 . 4 5 3 4 E - 03 0 . 2 5 6 2 E - 03 0 . 7 4 6 2 E - 04 0 . 2 6 3 0 E - 04 0.1136E- 04 0 . 5 6 7 4 E - 05 0 . 2 0 4 0 E - 05 0 . 1 5 6 0 E - 05 0 . 4 3 5 1 E - 06 0 . 3 6 3 2 E - 06
0 . 2 9 0 9 E - 01 0 . 4 8 8 7 E - 01 0 . 1 3 8 6 E + 00 0 . 6 6 8 3 E + 00 0 . 2 1 1 4 E + 01 0 . 8 7 9 0 E + 00 0.1389E+00 0 . 7 2 7 7 E - 01 0 . 1 6 3 9 E - 01 0 . 7 3 6 8 E - 02 0 . 2 6 0 9 E - 02 0 . 1 6 1 3 E - 02 0 . 4 5 1 4 E - 03 0 . 2 5 5 0 E - 03 0.7421E - 04 0.2612E - 04 0.1129E- 04 0 . 5 6 3 4 E - 05 0 . 2 0 2 7 E - 05 0 . 1 5 4 8 E - 05 0 . 4 3 2 6 E - 06 0 . 3 6 0 5 E - 06
0.2907E0.4892E = 0.1386E + 0.6683E + 0.2114E + 0.8791E + 0.1389E+ 0.7283E0.1635E0.7410E 0.2566E 0.1655E 0.4205E0.2889E0.5760E 0.6478E -
01 01 00 00 01 00 00 01 01 02 02 02 03 03 04 04
Table 3 T h e a v e r a g e d v e c t o r (VT) m e s o n p r o p a g a t o r m e a s u r e d u s i n g f o u r c o n f i g u r a t i o n s o n a 163 X 24 lattice at fl = 6.0 using, in c o l u m n 1 the D S M , in c o l u m n 2 the iterative B l o c k S O R a l g o r i t h m a n d in c o l u m n 3 the c o n j u g a t e g r a d i e n t a l g o r i t h m o n a 164 lattice. T h e q u a r k m a s s is 0.09. Time
DSM vector
IBSOR vector
C G 164 v e c t o r
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0 . 1 8 6 6 E - 01 0 . 4 1 7 5 E - 01 0 . 1 1 8 9 E + 00 0 . 8 7 6 4 E + 00 0 . 1 7 8 8 E + 01 0.8116E+00 0 . 1 0 7 5 E + 00 0 . 5 0 6 7 E - 01 0 . 1 0 6 5 E - 01 0 . 4 0 8 7 E - 02 0 . 1 4 4 5 E - 02 0 . 7 4 5 1 E - 03 0 . 2 3 9 1 E - 03 0.1135E-03 0 . 3 6 9 4 E - 04 0 . 1 3 9 9 E - 04 0 . 5 8 4 7 E - 05 0 . 2 7 7 8 E - 05 0 . 1 0 0 9 E - 05 0 . 6 5 9 5 E - 06 0 . 1 2 3 0 E - 06 0 . 1 5 4 8 E - 06 0 . 3 5 9 6 E - 07 0 . 5 4 7 5 E - 07
0 . 1 8 6 7 E - 01 0 . 4 1 7 3 E - 01 0.1189E + 00 0.8764E + 00 0 . 1 7 8 8 E + 01 0.8115E+00 0.1075E + 00 0 . 5 0 6 4 E - 01 0 . 1 0 6 6 E - 01 0 . 4 1 9 8 E - 02 0 . 1 4 3 5 E - 02 0 . 7 4 3 1 E - 03 0 . 2 3 8 2 E - 03 0.1130E-03 0.3672E- 04 0.1390E- 04 0 . 5 8 1 2 E - 05 0 . 2 7 5 9 E - 05 0 . 1 0 0 1 E - 05 0 . 6 5 3 9 E - 06 0 . 1 2 1 9 E - 06 0 . 1 5 3 6 E - 06 0 . 3 5 6 5 E - 07 0 . 5 4 3 3 E - 07
0.1866E0.4175E0.1189E+ 0.8764E + 0.1788E + 0.8116E+ 0.1075E + 0.5067E 0.1065E0.4222E 0.1416E0.7603E0.2254E0.1315E0.2669E0.3004E -
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January1987
Table 4 A s table 3 but for a quark mass of 0.04 and including in column 4 the standard deviation in the IBSOR data. Time
D S M vector
IBSOR vector
C G 16 4
IBSOR
SDEV
1
0 . 2 2 8 5 D - 01
0 . 2 2 9 1 D - 01
0 . 2 2 8 5 D - 01
0.1271D - 01
2
0 . 4 2 8 5 D - 01
0 . 4 2 8 2 D - 01
0 . 4 2 8 5 D - 01
0.8747D - 02
3
0.1208D + 00
0.1209D + 00
0.1208D + 00
0 . 2 5 6 6 D - 01
4
0.8937D + 00
0.8940D + 00
0.8937D + 00
0.1980D +
5
0 . 1 7 0 8 D + 01
0 . 1 7 0 8 D + 01
0 . 1 7 0 8 D + 01
0.2822D +
6
0.8625D + 00
0.8619D + 00
0.8625D + 00
0.1226D + 00
7
0.1114D + 00
0.1116D + 00
0.1114D + 00
0.2681D - 01
8
0 . 6 3 2 7 D - 01
0 . 6 2 8 7 D - 01
0 . 6 3 2 7 D - 01
0 . 1 9 8 9 D - 01
9
0 . 1 4 5 3 D - 01
0 . 1 4 6 7 D - 01
0 . 1 4 5 3 D - 01
0.5356D - 02
10
0.5745D - 02
0.6915D - 02
0.7179D - 02
0.4284D - 02
11
0.3010D-
02
0.2584D - 02
0.2427D - 02
0.1711D - 02
12
0.2014D - 02
0.1957D - 02
0.2109D - 02
0.1836D - 02
13
0.7189D - 03
0.6937D - 03
0.5986D - 03
0.7021D - 03
14
0.3570D - 03
0.3446D - 03
0.4782D - 03
0.4497D - 03
15
0.1700D - 03
0 . 1 6 1 9 D - 03
0 . 1 0 7 8 D - 03
0.1913D - 03
16
0.4017D - 04
0.3825D - 04
0.7265D - 04
0.9151D - 04
17
0.1043D - 04
0.1022D - 04
0.8322D - 04
18
0.1300D - 04
0.1204D - 04
0.4104D - 04
19
- 0.3278D - 05
- 0 . 3 7 8 3 D - 05
0.2782D - 04
20
0.1233D - 04
0.1123D - 04
0.1695D - 04
21
- 0.8109D-
0.7149D-
00 00
07
- 0.2133D - 06
22
0.6788D - 05
0.6160D - 05
0.7709D - 05
05
23
0.1689D - 05
0.1566D - 05
0.1745D - 05
24
0.4467D - 05
0.4081D - 05
0.4173D - 05
are worse. Table 3 shows the averaged vector propagator obtained by the two methods at m = 0.09 (from the same four configurations) and as we might expect, the two sets of data are very close. We have chosen to display the vector meson propagator because it is sensitive to cancellations in the statistical average, and hence is more representative than the pseudoscalar propagator. What is apparent from table 4 is that agreement is not quite so good at lower values of the quark mass, although, as can be seen from the column of standard deviations, the discrepancy between the two methods on almost all timeslices is at least an order of magnitude smaller than the statistical error. A probable explanation for the disagreement is that the second "source" T x 8 becomes increasingly contaminated by finite time effects as the quark mass is lowered, due to the small lattice only having 16 timeslices. At a given quark mass, the discrepancy between D S M and the IBSOR results is most marked near the "join" in the D S M i.e. on timeslices 10 and 11 (as can be seen clearly in table 4). Notice that, because we
Table 5 Numbers of iterations used for a two-stage D S M calculation of Mass
a 163 × 24
quark propagator at various quark masses
m.
Number of iterations on first (164 ) stage
on second stage
0.50
40
60+
30
0.16
120
100+
50
0.09
150
150+
50
0.04
300
250 +
50
0.01
500 + 200
600 + 300
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used the e v e n / o d d partitioned conjugate gradient algorithm [2], this effect sets in on timeslice 10 rather than on timeslice 9 which is the actual time boundary on the second, because timeslice 9 has the opposite " p a r i t y " to timeslices 8 and 10. The effect of this join becomes unobservable after two timeslices in our data. However, in high statistics measurements and at low quark masses, it m a y be sufficiently pronounced to require corrective action. We propose to replace the bad D S M timeslices (10 and 11) by the corresponding timeslices obtained for the first 164 lattice. In this latter data, the effect of timeslice 16 is relatively insignificant whenever finite-size effects are small, as is evident from table 4. To summarise then, we can see that the D S M works well as long as we do not lower the quark mass too far or, in general, provided finite-size effects are small. At the lower quark masses it might seem advisable to use three steps on a 164 lattice go that, say, x 6 could be used as the source in the second stage, and x12 in the third stage, to reduce finite-time effects. However, the method would probably then no longer be cost-effective: The results presented here demonstrate the accuracy of the D S M sufficiently clearly that we see no reason to extend the comparison with IBSOR to a larger statistical sample. The statistical error in state-of-the-art hadron propagator calculation is, of course, smaller than that in table 4. However, our results indicate that the systematic error in the D S M is only likely to become comparable with this in regimes where finite-size effects are already overwhelming. So far, the IBSOR has proved the more efficient way of obtaining 163 × 24 propagators on the D A P from scratch, and because these lattices appear to be long enough for our present analysis [4] (at least to the extent that signal-noise ratio problems begin around timesfice 19) there is no reason to extend these propagators. However, in future work with higher statistics it might be beneficial to extend the propagators in time to 163× 32 for example, permitting the inclusion of more timeslices in the fits to the asymptotic decay. Given the existence of a large set of 163 × 24 propagators, the D S M would be a more efficient way of accomplishing this than starting again from scratch. This work was done using the DAPs at Edinburgh which are supported by SERC grant N G 15 908. C.B.C. is supported by a Vans Dunlop Scholarshi p f r o m the University of Edinburgh and D.R. is supported by the SERC and Inmos Ltd.
References [1] R.D. Kenway, Phys. Lett B 158 (1985) 327. [2] C.B. Chalmers, R.D. Kenway and D. Roweth, Algorithms for calculating quark propagators on large lattices, Edinburgh preprint 86/361 (revised version), submitted to J. Comput. Phys. [3] N. Cabibbo and E. Marinari, Phys. Lett B 119 (1982) 387. [4] K.C. Bowler, C.B. Chalmers, R.D. Kenway, G.S. Pawley and D. Roweth, Hadron mass calculations using Susskind fermions at fl = 5.7 and 6.0, Edinburgh preprint 86/369, Nucl. Phys. B., to be published.
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