2-17 PARALLEL AND VECTORIZED FORTRAN
*************************************
Using vector and parallel computers require somewhat similar
considerations. In simple words, both types of computers try to execute
more than one FORTRAN statement at the same time, and thus change the
statement execution order (see the examples below). We must ensure that
changing the execution order of statements will not change the results.
The problem of vectorizing/parallelizing non-looping code is difficult,
and vectorizing/parallelizing is usually limited to DO loops that satisfy
a condition of DATA INDEPENDENCE.
Vector machines have longer registers that can hold at the same time
several data items (up to 128) and operate on them. For example,
the register may be loaded with data items that belong to some
consecutive iterations of a DO loop, thus more than one iteration
can be performed at the same time.
Parallel machines can execute different iterations of a DO loop on
different processors at the same time. The processors may have shared
main memory or they can be independent and connected by a fast network.
By the way, the term 'vector' is used here in a similar way to that
used in linear algebra, it means a sequence of scalar data items.
Bernstein conditions for single loop independence
-------------------------------------------------
We will assume that variables or array elements appearing in
the loop aquire their values only by assignment statements.
Consider every iteration of the loop as a separate unit, with
the variables and array elements that are referenced in it.
A variable or array element appearing on the right side of an
assignment statement will be called an INPUT VARIABLE of the
iteration it belongs to.
Similarly a variable or array element appearing on the left
side of an assignment will be called an OUTPUT VARIABLE of
the iteration it belongs to.
In C terminology we call an input variable a rvalue, and an
output variable a lvalue.
If the following two conditions are true, NO DATA DEPENDENCY
IS POSSIBLE between iterations:
1) No output variable is also an input variable
in another iteration.
2) No output variable is also an output variable
in another iteration.
If this simple criterion is violated in a loop, we have to
perform some kind of analysis (see below) in order to determine
if a dependency really exists, and whether it is a REMOVABLE.
The variables that violated the criterion will help us classify
the type of dependency involved.
Remark: Auxiliary variables, that can be eliminated algebraically,
and are computed before they are used, are output variables
according to our definition, but can be safely ignored.
Vectorizing/Parallelizing compilers
-----------------------------------
Vectorizing/Parallelizing compiler technology is quite developed,
loops are analyzed automatically for data dependencies and if found
independent the compiler generates vector code.
Compilers use algorithms like the Banerjee's inequalities or the
GCD tests to analyze dependencies.
The compiler will warn you of dependencies, and you should go over
them armed with the concepts discussed in this chapter, and try to
remove them.
Classification of dependencies
------------------------------
The dependencies we will discuss are:
Store before load Competing stores Load before store
================= ================ ===============
DO I=2,4 DO I=2,4 DO I=2,4
A(I-1) = B(I) A(I-1) = B(I) B(I) = A(I-1)
C(I) = A(I) A(I) = C(I) A(I) = C(I)
ENDDO ENDDO ENDDO
Recurrence Sum reduction
=============== ==============
DO I=2,4 SUM = 0.0
A(I) = A(I-1) DO I=2,4
ENDDO SUM = SUM + A(I)
ENDDO
Removable data dependancy: store before load
--------------------------------------------
A small example:
DO I=2,4
A(I-1) = B(I)
C(I) = A(I)
ENDDO
Let's unroll the loop and see the correct order for executing
the assignment statements:
t=1 A(1) = B(2)
t=2 C(2) = A(2)
t=3 A(2) = B(3)
t=4 C(3) = A(3)
t=5 A(3) = B(4)
t=6 C(4) = A(4)
Performing all iterations in parallel:
Parallel #1 Parallel #2 Parallel #3
=========== =========== ===========
t=1 A(1) = B(2) A(2) = B(3) A(3) = B(4)
t=2 C(2) = A(2) C(3) = A(3) C(4) = A(4)
We see that the execution order is changed when running the
loop in parallel, e.g. the first assignment of #2 (t=3) comes
before the second assignment of #1 (t=2).
When the second assignment of #1 executes, the value of A(2)
is already corrupted, the first assignment of #2 gave it the
value of B(3) instead of its original value. The same problem
occurs between #2 and #3, the data is overwritten before
being used and the result is wrong.
A simple modification can save the parallel code:
DO I=2,4
TEMP = A(I)
A(I-1) = B(I)
C(I) = TEMP
ENDDO
This technique is called PRE-LOADING, the endangered value
of A(I) is temporarily stored in the new auxiliary variable
TEMP, the 'protective store' is done at the top of the loop.
Possibly removable data dependancy: two competing stores
--------------------------------------------------------
A small example:
DO I=2,4
A(I-1) = B(I)
A(I) = C(I)
ENDDO
Unrolled:
t=1 A(1) = B(2)
t=2 A(2) = C(2)
t=3 A(2) = B(3)
t=4 A(3) = C(3)
t=5 A(3) = B(4)
t=6 A(4) = C(4)
Parallelized:
Parallel #1 Parallel #2 Parallel #3
=========== =========== ===========
t=1 A(1) = B(2) A(2) = B(3) A(3) = B(4)
t=2 A(2) = C(2) A(3) = C(3) A(4) = C(4)
We see that the execution order is changed when running the
loop in parallel, e.g. the first assignment of #2 (t=3) comes
before the second assignment of #1 (t=2).
The value left in A(2) after the loop executes is wrong,
instead of B(3) it is equal to C(2). The same problem occurs
between #2 and #3, the order of the store operations is wrong
and the result is wrong.
Again a simple modification can save the parallel code,
we just SWITCH LINES inside the loop:
DO I=2,4
A(I) = C(I)
A(I-1) = B(I)
ENDDO
This simple trick will make the order of stores come right,
IF we can assume that the parallel 'tracks' are synchronous.
Vector processors are synchronous, parallel machines may
be not (?).
Possibly removable data dependancy: load before store
-----------------------------------------------------
A small example:
DO I=2,4
B(I) = A(I-1)
A(I) = C(I)
ENDDO
Unrolled:
t=1 B(2) = A(1)
t=2 A(2) = C(2)
t=3 B(3) = A(2)
t=4 A(3) = C(3)
t=5 B(4) = A(3)
t=6 A(4) = C(4)
Parallelized:
Parallel #1 Parallel #2 Parallel #3
=========== =========== ===========
t=1 B(2) = A(1) B(3) = A(2) B(4) = A(3)
t=2 A(2) = C(2) A(3) = C(3) A(4) = C(4)
We see that the execution order is changed when running the
loop in parallel, e.g. the first assignment of #2 (t=3) comes
before the second assignment of #1 (t=2).
The value of A(2) is used before it is assigned the right
value C(2). The same problem occurs between #2 and #3,
the variable is assigned before it contains the right value.
The order of the store operations is wrong and the result
is wrong.
Again a simple modification can save the parallel code,
looking closely in the code we can see that SWITCHING
LINES inside the loop will not change the result, and
will solve the dependency problem:
DO I=2,4
A(I) = C(I)
B(I) = A(I-1)
ENDDO
This simple trick will make the order of stores come right,
IF we can assume that the parallel 'tracks' are synchronous.
Vector processors are synchronous, parallel machines may
be not (?).
Irremovable data dependency: recurrence
-----------------------------------------
A small example:
DO I=2,4
A(I) = A(I-1)
ENDDO
Unrolled:
t=1 A(2) = A(1)
t=2 A(3) = A(2)
t=3 A(4) = A(3)
Parallelized:
Parallel #1 Parallel #2 Parallel #3
=========== =========== ===========
t=1 A(2) = A(1) A(3) = A(2) A(4) = A(3)
Well, nothing can be done in this case, the dependency is essential,
and can't be removed by some simple manipulation.
Bernstein conditions for double loop independence
-------------------------------------------------
If we have either a vector machine or a parallel one, treating 2 level
loop nests is simple, we just vectorize/parallelize one of the loops.
With a machine that is both vector and parallel we would try to vectorize
the inner loop and parallelize the outer one.
Dependency analysis is more complicated in this case ....
Other loops: FORALL, DOALL
--------------------------
Shared memory systems
---------------------
... interleaved memory ...
Memory contention
-----------------
Distributed memory systems and message passing
----------------------------------------------
Bibliography
------------
An nice introduction to hardware and software:
Advanced Computer Architecture:
Parallelism, Scalability, Programmability
Kai Hwang
McGraw-Hill 1993
Library og Congress QA76.9 A73 H87
ISBN 0-07-113342-9
A much more important factor in the social movement than those already mentioned was the ever-increasing influence of women. This probably stood at the lowest point to which it has ever fallen, during the classic age of Greek life and thought. In the history of Thucydides, so far as it forms a connected series of events, four times only during a period of nearly seventy years does a woman cross the scene. In each instance her apparition only lasts for a moment. In three of the four instances she is a queen or a princess, and belongs either to the half-barbarous kingdoms of northern Hellas or to wholly barbarous Thrace. In the one remaining instance208— that of the woman who helps some of the trapped Thebans to make their escape from Plataea—while her deed of mercy will live for ever, her name is for ever lost.319 But no sooner did philosophy abandon physics for ethics and religion than the importance of those subjects to women was perceived, first by Socrates, and after him by Xenophon and Plato. Women are said to have attended Plato’s lectures disguised as men. Women formed part of the circle which gathered round Epicurus in his suburban retreat. Others aspired not only to learn but to teach. Arêtê, the daughter of Aristippus, handed on the Cyrenaic doctrine to her son, the younger Aristippus. Hipparchia, the wife of Crates the Cynic, earned a place among the representatives of his school. But all these were exceptions; some of them belonged to the class of Hetaerae; and philosophy, although it might address itself to them, remained unaffected by their influence. The case was widely different in Rome, where women were far more highly honoured than in Greece;320 and even if the prominent part assigned to them in the legendary history of the city be a proof, among others, of its untrustworthiness, still that such stories should be thought worth inventing and preserving is an indirect proof of the extent to which feminine influence prevailed. With the loss of political liberty, their importance, as always happens at such a conjuncture, was considerably increased. Under a personal government there is far more scope for intrigue than where law is king; and as intriguers women are at least the209 equals of men. Moreover, they profited fully by the levelling tendencies of the age. One great service of the imperial jurisconsults was to remove some of the disabilities under which women formerly suffered. According to the old law, they were placed under male guardianship through their whole life, but this restraint was first reduced to a legal fiction by compelling the guardian to do what they wished, and at last it was entirely abolished. Their powers both of inheritance and bequest were extended; they frequently possessed immense wealth; and their wealth was sometimes expended for purposes of public munificence. Their social freedom seems to have been unlimited, and they formed combinations among themselves which probably served to increase their general influence.321 The old religions of Greece and Italy were essentially oracular. While inculcating the existence of supernatural beings, and prescribing the modes according to which such beings were to be worshipped, they paid most attention to the interpretation of the signs by which either future events in general, or the consequences of particular actions, were supposed to be divinely revealed. Of these intimations, some were given to the whole world, so that he who ran might read, others were reserved for certain favoured localities, and only communicated through the appointed ministers of the god. The Delphic oracle in particular enjoyed an enormous reputation both among Greeks and barbarians for guidance afforded under the latter conditions; and during a considerable period it may even be said to have directed the course of Hellenic civilisation. It was also under this form that supernatural religion suffered most injury from the great intellectual movement which followed the Persian wars. Men who had learned to study the constant sequences of Nature for themselves, and to shape their conduct according to fixed principles of prudence or of justice, either thought it irreverent to trouble the god about questions on which they were competent to form an opinion for themselves, or did not choose to place a well-considered scheme at the mercy of his possibly interested responses. That such a revolution occurred about the middle of the fifth century B.C., seems proved by the great change of tone in reference to this subject which one perceives on passing from Aeschylus to Sophocles. That anyone should question the veracity of an oracle is a supposition which never crosses the mind of the elder dramatist. A knowledge of augury counts among the greatest benefits222 conferred by Prometheus on mankind, and the Titan brings Zeus himself to terms by his acquaintance with the secrets of destiny. Sophocles, on the other hand, evidently has to deal with a sceptical generation, despising prophecies and needing to be warned of the fearful consequences brought about by neglecting their injunctions. The stranger had a pleasant, round face, with eyes that twinkled in spite of the creases around them that showed worry. No wonder he was worried, Sandy thought: having deserted the craft they had foiled in its attempt to get the gems, the man had returned from some short foray to discover his craft replaced by another. “Thanks,” Dick retorted, without smiling. When they reached him, in the dying glow of the flashlight Dick trained on a body lying in a heap, they identified the man who had been warned by his gypsy fortune teller to “look out for a hidden enemy.” He was lying at full length in the mould and leaves. "But that is sport," she answered carelessly. On the retirement of Townshend, Walpole reigned supreme and without a rival in the Cabinet. Henry Pelham was made Secretary at War; Compton Earl of Wilmington Privy Seal. He left foreign affairs chiefly to Stanhope, now Lord Harrington, and to the Duke of Newcastle, impressing on them by all means to avoid quarrels with foreign Powers, and maintain the blessings of peace. With all the faults of Walpole, this was the praise of his political system, which system, on the meeting of Parliament in the spring of 1731, was violently attacked by Wyndham and Pulteney, on the plea that we were making ruinous treaties, and sacrificing British interests, in order to benefit Hanover, the eternal millstone round the neck of England. Pulteney and Bolingbroke carried the same attack into the pages of The Craftsman, but they failed to move Walpole, or to shake his power. The English Government, instead of treating Wilkes with a dignified indifference, was weak enough to show how deeply it was touched by him, dismissed him from his commission of Colonel of the Buckinghamshire Militia, and treated Lord Temple as an abettor of his, by depriving him of the Lord-Lieutenancy of the same county, and striking his name from the list of Privy Councillors, giving the Lord-Lieutenancy to Dashwood, now Lord Le Despencer. "I tell you what I'll do," said the Deacon, after a little consideration. "I feel as if both Si and you kin stand a little more'n you had yesterday. I'll cook two to-day. We'll send a big cupful over to Capt. McGillicuddy. That'll leave us two for to-morrer. After that we'll have to trust to Providence." "Indeed you won't," said the Surgeon decisively. "You'll go straight home, and stay there until you are well. You won't be fit for duty for at least a month yet, if then. If you went out into camp now you would have a relapse, and be dead inside of a week. The country between here and Chattanooga is dotted with the graves of men who have been sent back to the front too soon." "Adone do wud that—though you sound more as if you wur in a black temper wud me than as if you pitied me." "Wot about this gal he's married?" "Don't come any further." "Davy, it 'ud be cruel of us to go and leave him." "Insolent priest!" interrupted De Boteler, "do you dare to justify what you have done? Now, by my faith, if you had with proper humility acknowledged your fault and sued for pardon—pardon you should have had. But now, you leave this castle instantly. I will teach you that De Boteler will yet be master of his own house, and his own vassals. And here I swear (and the baron of Sudley uttered an imprecation) that, for your meddling knavery, no priest or monk shall ever again abide here. If the varlets want to shrieve, they can go to the Abbey; and if they want to hear mass, a priest can come from Winchcombe. But never shall another of your meddling fraternity abide at Sudley while Roland de Boteler is its lord." "My lord," said Edith, in her defence, "this woman has sworn falsely. The medicine I gave was a sovereign remedy, if given as I ordered. Ten drops would have saved the child's life; but the contents of the phial destroyed it. The words I uttered were prayers for the life of the child. My children, and all who know me, can bear witness that I have a custom of asking His blessing upon all I take in hand. I raised my eyes towards heaven, and muttered words; but, my lord, they were words of prayer—and I looked up as I prayed, to the footstool of the Lord. But it is in vain to contend: the malice of the wicked will triumph, and Edith Holgrave, who even in thought never harmed one of God's creatures, must be sacrificed to cover the guilt, or hide the thoughtlessness of another." "Aye, Sir Treasurer, thou hast reason to sink thy head! Thy odious poll-tax has mingled vengeance—nay, blood—with the cry of the bond." HoME古一级毛片免费观看
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