Performance Tables

Table 4.1: Computer used for running the performance timing

Computer	Processor	OS	Compiler	BLAS
name	name	version	version & options	library
COMPAQ	Alpha EV6	OSF1 - Tru64	f90 v. 5.3	CXML
	@ 500 MHz	v. 4.0	Compaq Fortran	v. 3.5
			-O3
IBM	PowerPC 604e	AIX	xlf95 v. 6.1	ESSL
	@ 332 MHz	v. 4.3.3	IBM compiler	v. 3.1.1
			-O3 -qstrict
IBM	Power2	AIX	xlf95 v. 6.1	ESSL
	@ 67 MHz	v. 4.3.3	IBM compiler	v.3.1.1
			-O3 -qstrict
			-qarch=pwr2
SUN	UltraSparc II	SUNOS	f90 v. 5.0	SUNPERF
	@ 400 MHz	v. 5.7	Workshop compiler	v. 2.0
			-fast -xtarget=ultra2
			-xarch=v8plusa
INTEL	Pentium III	Linux RedHat	f95 v. 1.0	ATLAS
	@ 500 MHz	v. 6.1	NAG compiler	v. 3.0
			-O3
SGI	R12000	IRIX64	f90 v. 7.30	ATLAS
	@ 300 MHz	v.6.5	MIPSpro compiler	v. 3.0
			-O3

In this section we give performance results, in megaflops, for some basic computations on a variety of computers. Table 4.1 lists the computers used, processor names, operating system versions, compiler versions and the BLAS library versions. Regarding the latter, ATLAS refers to BLAS obtained from the ATLAS system; see Section 1.5.3. ESSL is the Engineering Scientific Subroutine Library [23,22] for IBM computers; it also contains the IBM specialized BLAS. SUNPERF is the Sun WorkShop(TM) 6 Performance Library [41]; it also contains the SUN specialized BLAS. CXML is the Compaq Extended Math Library [7]; it also contains the Compaq Alpha specialized BLAS.
Each of the performance tables below gives the performance of a specific LAPACK95 driver routine and, in addition, the performance obtained by using LAPACK directly; i.e., without the LAPACK95 interface. A table is arranged as follows: Column 1 identifies the computers and their processors. Column 2 gives the optimal block size (Section 1.5.1.1). Column 3 specifies the type of data: D, real double precision; S, real single precision; Z, complex double precision; C, complex single precision. Columns 4 and 5 give the megaflop counts achieved by LAPACK, without the Fortran 95 interface, for problems of order 100 and 1000, respectively. Columns 6 and 7 give the megaflop counts for the same problems when the LAPACK95 driver routine named in the figure caption is used.
Each of the megaflop counts in the tables was obtained as follows: The problem was run 10 times, and each time the elapsed time was measured using the Fortran 95 command CPU_TIME. The megaflop rate was then computed from the formula

$$Mflop\index{megaflops} = \frac{\alpha\times n^3}{t \times 10^6}$$

where $t$ is the average of the 10 elapsed times and $\alpha$ is given in Table 4.2.

Table: Floating point coefficient of operation counts for LAPACK drivers for $n\times n$ matrices (see also Table 3.13 of [1]). The number of operations is $\alpha \times n^3$.

Driver	Options	$\alpha$
LA_GESV	1 right hand side	0.67
LA_GEEV	eigenvalues only	10.00
LA_GEEV	eigenvalues and right eigenvectors	26.33
LA_GES{VD,DD}	singular values only	2.67
LA_GES{VD,DD}	singular values, and left and right singular vectors	6.67

The driver routines timed are:

• LA_GESV (Table 4.3): computes the solution to a system of equations AX = B, where A is an $n\times n$ matrix and B (in this computation) is an $n\times 1$ vector (Section 5.1.1).
  
• LA_GEEV (Tables 4.4 and 4.5): computes for an $n\times n$ matrix A the eigenvalues and, optionally, the left and/or right eigenvectors (Section 7.2.3).
  
• LA_GESVD (Table 4.6): computes for an $m \times n$ matrix A the singular values and, optionally, the left and/or right singular vectors (Section 9.1.1). In this computation $m = n$.
  
• LA_GESDD (Tables 4.7 and 4.8): solves the same problem as LA_GESVD but uses a divide and conquer method if singular vectors are desired (Section 9.1.1).

Table: Performance of LA_GESV in megaflops; $n = 100$ and $1000$.

Computer /	Block	Data	LAPACK		LAPACK95
Processor	size	type	100	1000	100	1000
COMPAQ Alpha EV6	28	D	402	732	402	679
@ 500 MHz		S	402	789	402	755
		Z	80	152	80	151
		C	81	174	81	171
IBM PowerPC 604e	32	D	104	271	101	271
@ 332 MHz		S	145	333	145	333
		Z	57	243	57	226
		C	67	112	67	111
IBM Power2	32	D	67	236	67	235
@ 67 MHz		S	67	218	67	218
		Z	33	58	33	58
		C	33	60	33	59
SUN UltraSparc II	64	D	109	177	109	172
@ 400 MHz		S	130	247	155	249
		Z	40	35	37	35
		C	40	46	39	46
INTEL Pentium III	40	D	67	251	67	251
@ 500 MHz		S	67	314	67	314
		Z	34	71	34	71
		C	66	88	66	88
SGI R12000	64	D	182	442	190	445
@ 300 MHz		S	242	340	242	344
		Z	59	113	60	114
		C	65	127	66	127

Table: Performance of LA_GEEV in megaflops (eigenvalues only); $n = 100$ and $1000$.

Computer /	Block	Data	LAPACK		LAPACK95
Processor	size	type	100	1000	100	1000
COMPAQ Alpha EV6	28	D	272	262	270	263
@ 500 MHz		S	300	369	300	372
IBM PowerPC 604e	32	D	109	87	109	99
@ 332 MHz		S	117	128	117	161
IBM Power2	32	D	49	76	50	76
@ 67 MHz		S	51	94	51	98
SUN UltraSparc II	64	D	105	95	118	97
@ 400 MHz		S	152	176	171	173
INTEL Pentium III	40	D	95	97	91	93
@ 500 MHz		S	142	175	142	175
SGI R12000	64	D	171	236	177	233
@ 300 MHz		S	230	407	230	411

Table: Performance of LA_GEEV in megaflops (eigenvalues and right eigenvectors); $n = 100$ and $1000$.

Computer	Block	Data	LAPACK		LAPACK95
Processor	size	type	100	1000	100	1000
COMPAQ Alpha EV6	28	D	267	268	267	268
@ 500 MHz		S	376	437	351	437
IBM PowerPC 604e	32	D	141	81	141	85
@ 332 MHz		S	148	138	147	171
IBM Power2	32	D	69	70	68	70
@ 67 MHz		S	63	95	63	108
SUN UltraSparc II	64	D	104	92	100	99
@ 400 MHz		S	197	181	199	183
INTEL Pentium III	40	D	112	105	112	111
@ 500 MHz		S	181	201	181	207
SGI R12000	64	D	246	241	249	257
@ 300 MHz		S	325	500	325	479

Table: Performance of LA_GESVD in megaflops (singular values and left and right singular vectors); $n = 100$ and 1000.

Computer /	Block	Data	LAPACK		LAPACK95
Processor	size	type	100	1000	100	1000
COMPAQ Alpha EV6	28	D	130	60	129	60
@ 500 MHz		S	174	105	181	104
IBM PowerPC 604e	32	D	43	22	43	22
@ 332 MHz		S	56	29	56	29
IBM Power2	32	D	32	11	32	11
@ 67 MHz		S	32	17	32	16
SUN UltraSparc II	64	D	52	15	52	13
@ 400 MHz		S	65	33	64	32
INTEL Pentium III	40	D	49	31	49	31
@ 500 MHz		S	66	51	66	47
SGI R12000	64	D	90	42	90	42
@ 300 MHz		S	129	100	129	97

Table: Performance of LA_GESDD in megaflops (singular values only); $n = 100$ and $1000$.

Computer /	Block	Data	LAPACK		LAPACK95
Processor	size	type	100	1000	100	1000
COMPAQ Alpha EV6	28	D	267	300	267	293
@ 500 MHz		S	285	459	236	456
IBM PowerPC 604e	32	D	78	83	78	83
@ 332 MHz		S	110	119	110	119
IBM Power2	32	D	53	136	53	136
@ 67 MHz		S	56	134	66	138
SUN UltraSparc II	64	D	85	87	85	87
@ 400 MHz		S	140	150	119	144
INTEL Pentium III	40	D	89	121	89	121
@ 500 MHz		S	133	180	133	179
SGI R12000	64	D	134	280	134	280
@ 300 MHz		S	201	369	202	369

Table: Performance of LA_GESDD in megaflops (singular values and left and right singular vectors); $n = 100$ and $1000$.

Computer /	Block	Data	LAPACK		LAPACK95
Processor	size	type	100	1000	100	1000
COMPAQ Alpha EV6	28	D	210	372	200	355
@ 500 MHz		S	285	486	235	485
		C	88	108	88	107
		Z	100	135	80	132
IBM PowerPC 604e	32	D	74	118	74	118
@ 332 MHz		S	96	161	96	161
		Z	45	135	46	132
		C	50	91	50	92
IBM Power2	32	D	41	121	41	121
@ 67 MHz		S	43	124	43	124
		Z	23	45	23	45
		C	24	47	24	47
SUN UltraSparc II	64	D	81	86	81	77
@ 400 MHz		S	112	146	112	146
		Z	32	18	30	18
		C	35	37	34	36
INTEL Pentium III	40	D	78	146	74	146
@ 500 MHz		S	95	191	95	194
		Z	30	51	30	51
		C	39	65	39	65
SGI R12000	64	D	111	272	111	272
@ 300 MHz		S	169	311	170	311
		Z	50	77	52	78
		C	59	110	59	109