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Published
**1992** by Cornell Theory Center, Cornell University in Ithaca, N.Y .

Written in English

Read online**Edition Notes**

Statement | Wei Li, Keshav Pingali. |

Series | Technical report / Cornell Theory Center -- CTC92TR98., Technical report (Cornell Theory Center) -- 98. |

Contributions | Pingali, Keshav., Cornell Theory Center., Cornell Theory Center. Advanced Computing Research Institute., Workshop on Language and Compilers for Parallelism (5th : 1992 : New Haven, Conn.) |

The Physical Object | |
---|---|

Pagination | 22 p. : |

Number of Pages | 22 |

ID Numbers | |

Open Library | OL16958884M |

**Download singular loop transformation framework based on non-singular matrices**

Abstract. In this paper, we discuss a loop transformation framework that is based on integer non-singular matrices. The transformations included in this framework are called Λ-transformations and include permutation, skewing and reversal, as well as a transformation called loop framework is more general than existing ones; however, it is also more difficult to Cited by: In this paper, we discuss a loop transformation framework that is based on integer non-singular matrices.

The transformations included in this framework are called Λ-transformations and include permutation, skewing and reversal, as well as a transformation calledloop scaling. This framework is more general than existing ones; however, it is also more difficult to generate code in our by: CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): In this paper, we discuss a loop transformation framework that is based on integer non-singular matrices.

The transformations included in this framework are called -transformations and include permutation, skewing and reversal, as well as a transformation called loop scaling. Abstract. In this paper, we discuss a loop transformation framework that is based on integer non-singular matrices.

The transformations included in this framework are called -transformations and include permutation, skewing and reversal, as well as a transformation called loop : Wei Li and Keshav Pingali. In this paper, we discuss a loop transformation framework that is based on integer non-singular matrices.

The transformations included in this framework are called -transformations and include. matrices. Non-singular linear transformations presented here subsumes the class of unimodular transformations. The loop transformations included are the unimodular transformations -- reversal, skewing, permutation -- and a new transformation, namely stretching.

Non-unimodular transformations (with determinant) create. Access Normalisation: Loop Restructuring for NUMA Compilers Cornell University, Department of Computer Science, TRf8, April Google Scholar Digital Library; W.

Li and K. Pingali. A Singular Loop Transformation framework Based on Non-singular Matrices, Proc. of the Fifth Workshop on Languages and Compilers for Parallelism, August Wei Li and Keshav Pingali.

A singular loop transformation based on non-singular matrices. International Journal of Parallel Programming, 22(2), April Google Scholar Digital Library; William Pugh.

The omega test: A fast and practical integer programming algorithm for dependence analysis. In Communications of the ACM, pagesAugust.

Wei Li and Keshav Pingali. A singular loop transformation framework based on non-singular matrices. In Proceedings of the 5th International Workshop on Languages and Compilers for Parallel Computing, August Google Scholar Digital Library; William Pugh.

The Omega test: A practical algorithm for exact singular loop transformation framework based on non-singular matrices book dependence analysis. If you think of the matrix in terms of being a linear transformation on $\mathbb{R}^n$, then a nonsingular matrix has full rank.

A singular matrix diminishes rank. Once you diminish rank, there is no way back. Hence the product of any square matrix with a singuluar matrix is singular.

Second, we develop a framework based on non-singular matrices and integer lattice theory for the systematic development of loop transformations. Program transformations, such as loop restructuring, are critical to achieving high performance. A singular loop transformation framework based on non-singular matrices.

In 5th Workshop on Languages and Compilers for Parallel Computing, pages –, Yale University, August Google Scholar. Such a capability is required for a global locality optimization framework that applies both loop and data transformation to a sequence of loop nests for optimizing locality.

Our method finds a non-singular iteratio-space transformation matrix such that in a given loop nest spatial locality is exploited in the innermost loops where it is most. W. Li, K. Pingali, ”A singular loop transformation framework based on non-singular matrices”, In Languages and Compilers for Parallel Computing, Yale University, August We have developed a framework based on non-singular matrices for the systematic development of loop transformations [9].

The main benefit of this framework is that it provides an approach to tackling the so-called `phase-ordering problem' for many problems where there is no obvious order in which the transformations should be performed, it.

Journals & Books; Help Download PDF This paper is concerned with integrating global data transformations and local loop transformations in order to minimize overhead on distributed shared memory machines such as the SGi Origin K.

PingaliA singular loop transformation framework based on non-singular matrices. Internat. Parallel. We have developed a framework based on non-singular matrices for the systematic development of loop transformations [9]. The main benefit of this framework is that it provides an approach to tackling the so-called `phase-ordering problem' for many problems where there is no obvious order in which the transformations should be performed.

Singular Transformations and Matrices A linear transformation T from an n dimensional space to itself (or an n by n matrix) is singular when its determinant vanishes. This means that there is a linear combination of its columns (not all of whose coefficients are 0) which sums to the 0 vector.

non-singular square transformation matrix T. Such an invertible loop transformation matrix realizes the following transformation L I + o!L T 1 I 0 + o; where = is the new iteration vector (af-ter the transformation).

Similarly, for a m-dimensional array, an m non-singular data transformation matrix M has the follow-ing effect [24, 29, 17]: L I. Abstract. In this paper, we discuss a loop transformation framework that is based on integer non-singular matrices. The transformations included in this framework are called -transformations and include permutation, skewing and reversal, as well as a transformation called loop scaling.

Li and K. Pingali. A singular loop transformation framework based on non-singular matrices. Proc. 5th Workshop on Languages and Compilers for Parallel Computing, August 8 J. Ramanujam and P. Sadayappan. Compile-time techniques for data distribution in distributed memory machines.

Linear loop transformations framework. Lambda is a framework that allows transformations of loops using non-singular matrix based transformations of the iteration space and loop bounds. This allows compositions of skewing, scaling, interchange, and reversal transformations.

Such a capability is required for a global locality optimiza-tion framework that applies both loop and data transformations to a sequence of loop nests for optimizing locality.

Our method finds a non-singular iteration-space transformation matrix such that in a given loop nest spatial locality is exploited in the innermost loops where it is.

1) where A, B, C and D are matrix sub-blocks of arbitrary size. (A must be square, so that it can be inverted. Furthermore, A and D − CA −1 B must be nonsingular.) This strategy is particularly advantageous if A is diagonal and D − CA −1 B (the Schur complement of A) is a small matrix, since they are the only matrices requiring inversion.

This technique was reinvented several times. As matrices usually have nonzero determinant and such matrices have behave more nicely, matrices with zero determinants were considered exceptional, hence the term "singular.

Li and K. Pingali, A singular loop transformation framework based on non-singular matrices, In: U. Banerjee, D. Gelernter, A.

Nicolau and D. Padua, editors. Languages and Compilers for Parallel Computing, Lecture Notes in Computer Science (Springer-Verlag, ) [14]. An example of the eigenvalue and singular value decompositions of a small, square matrix is provided by one of the test matrices from the Matlab gallery.

A = gallery(3) The matrix is A = − −50 − −27 −9 − This matrix was constructed in such a way that the characteristic polynomial factors nicely: det(A−λI. In this lesson we talk about what happens when the transformation matrix is singular - no matter where your original point is on the plane, the image will form a straight line.

In a recent paper, regarding mathematical framework to define and analyze a general parametric form of an arbitrary non-singular Mueller matrix, we proposed to address this problem in a space with more than 4 dimensions in order to introduce a set of transformations with an adequate corresponding number of degrees of freedom (DOF) and a group.

Definition: A linear transformation is said to be singular if it's nullspace contains atleast one non zero vector. I read another definition of a closely related topic which included matrices which is as follows: A matrix is said to be singular if it's determinant is 0.

I tried relating the two. We show a simple and effective matrix-based framework to implement this process. The search space that we consider for possible loop transformations can be represented by general non-singular linear transformation matrices and the data layouts that we conside Year: OAI identifier: oai.

Linear loop transformations framework Lambda is a framework that allows transformations of loops using non-singular matrix based transformations of the iteration space and loop bounds.

This allows compositions of skewing, scaling, interchange, and reversal transformations. cess can be put in a simple matrix framework which can be manipu-lated by an optimizing compiler. The search space that we consider for possible loop transformations comprises general non-singular linear transformation matrices and the data layouts that we consider are those which can be expressed using hyperplanes.

Experiments. The insertion of the real time network in the feedback control loop has many advantages and meanwhile can also make the analysis and design of the systems more complicated.

Here the transformation matrix T is non-singular if A 22 L. Moreau, D. NesicAn unified framework for input-to-state stability in systems with two time scales. IEEE. range of basic loop transformations (including loop interchange, loop reversal, loop skewing [Bane91] and loop scaling [LiPi92]).

In fact, any linear transformation modelled with a non-singular transformation matrix can be seen as a composition or product of these four basic transformations.

Liansheng Tan, in A Generalized Framework of Linear Multivariable Control, Abstract. The finite and infinite frequency structures of a rational matrix are fundamental to system analysis and design.

The classical methods of determining them are not stable in numerical computations, because the methods are based on unimodular matrix transformations, which result in an extraordinarily large. non-singular square matrices. Such a transformation matrix is invertible and is ofsizen× nfor an n-dimensional loop nest.

The eﬀect ofsuch a transforma-tion T is that each iteration vector I¯ in the original loop nest is transformed to T I¯. Therefore, loop bounds and subscript expressions should be modiﬁed accordingly.

Episode 04 of the video lectures on chapter 04 of the Mathematics textbook for class 12; covers singular and nonsingular matrices and solutions to problems. Loop permutation must usually be combined with other loop transformations like skewing, reversal and scaling for maximum advantage.

We have invented a loop transformation framework based on integer non-singular matrices. This framework synthesizes desirable sequences of loop transformations to enahnce parallelism and locality of reference.

If A is an n-square matrix it can be viewed as mapping n-vectors into n-vectors i.e. mapping n-space into itself. Matrix A may be singular or non-singular.

The mapping has an inverse if and only if the matrix A is non-singular. The inputs that matrix A operates on can be viewed as vectors or as points. It is just a matter of point of view, of. This paper describes transformation techniques for out-of-core pro-grams (i.e., those that deal with very large quantities of data) based on exploiting locality using a combination of loop and data trans-formations.

Writing efﬁcient out-of-core program is an arduous task. As a result, compiler optimizations directed at improving.abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix kernel linear.A singular linear operator (or matrix) is one whose determinant is zero.

Thm(half of linear algebra): Let A be a square, nxn matrix. The following are equivalent: 1)A is singular 2)A is not invertible 3)The kernel of A has dimension greater than z.