T. F. Coleman, B. S. Garbow and J. J. Mor'e [1984], "Software for estimating sparse Jacobian matrices", ACM Trans. Math. Software 11, pp. 363-378.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(k 1) im Pi(Pi 1) where Pi is the number of nonzero in the ith row of A. In practice, however, the numbers n and m are smaller due to the presence of segmented columns consisting of only zeros. In [18] we have used the exact coloring algorithm DSATUR [1] and a column ordering algorithm DSM [3]. Our test suite consisted of matrices from HarwellBoeing test collection [10] DSM have shown no improvements (in terms of the number of colors required) at all on the extended graph Gn(A) over G(A) for all the test instances. On the other hand, on some instances DSATUR required fewer colors on ....

T. F. Coleman, B. S. Garbow, and J. J. Mor. Software for estimating sparse Jacobian matrices. ACM Trans. Math. Software, 10(3):329-345, 1984.

....matrices V and W such that the nonzero elements of J can easily be extracted from the calculated pair (W T J, JV ) The motivation for solving above problem comes from the following two observations on the problem of computing a sparse Jacobian matrix. Sparse finitedi #erencing literature [CPR74,CGM84,CM84a,CM84b,CGM85,CC86] provides a solution based on partitioning of columns, to define a matrix V such that J can be determined from the product JV . However, the matrix V is not guaranteed to be thin, even if J has a lot of sparsity: consider a sparse matrix J with a single dense row. ....

....The bi partition problems can also be expressed in terms of graphs and graph coloring. This graph view is important in that it more readily exposes the relationship of the bi partition problems with the combinatorial approaches used in the sparse finite di#erencing literature, e.g. CPR74,CGM84,CM84a,CM84b,CGM85, CC86] 2.1.2 Algorithms for direct and substitution bi coloring The two combinatorial problems we face, corresponding to direct determination and determination by substitution, can both be approached in the following way. First, permute and partition the structure of J : J = ....

Thomas F. Coleman, Burton S. Garbow, and Jorge J. More. Software for estimating sparse Jacobian matrices. ACM Trans. Math. Software, 10(3):329--345, 1984.

....this section, we describe the computational experiments for the substitution schemes of 3 and 4. Our test problems are drawn from the HarwellBoeing [8] test matrix collection. For each of the test problems, we compute a row partition, a column partition, and a complete direct cover. We use DSM [4] to compute one directional partitioning and the algorithm in [12] to compute the direct cover. From the sparsity and partition information of the Jacobian matrix, the compressed matrix pattern is constructed. For each row of the pattern matrix we calculate the maximum number of consecutive zeros. ....

Thomas F. Coleman, Burton S. Garbow, and Jorge J. More. Software for estimating sparse Jacobian matrices. ACM Trans. Math. Software, 10(3):329--345, 1984.

....of the nonzero elements are calculated analytically using a user supplied subroutine, and the remainder of the elements are calculated using numerical finite di#erences. Further, the number of residual evaluations required for the finite di#erences is minimized through use of a special algorithm [8, 9]. modifications to allow use of DSL48S as part of a code to solve a broad class of high index DAEs [10] However, in itself DSL48S cannot solve high index DAEs. the option to use a novel algorithm developed at MIT that will automatically scale the matrix used in the corrector iteration ....

T.F. Coleman, B. S. Garbow, and J. J. More. Software for estimating sparse Jacobian matrices. ACM Trans. Math. Soft., 10(3):329--345, 1984.

....that the difference approximation to the derivatives is reliable but large enough so that rounding errors in the differences (8) are not important. 3 In many problems, J(x) is a sparse matrix, whose sparsity pattern is known. In this case, a procedure given in [20] and refined in [18] see also [17, 58]) allows one to compute a finite difference approximation to J(x) using less than n auxiliary functional evaluations. When the Jacobian matrix is dense, the discrete Newton method is not competitive with the cheaplinear algebra versions of (7) But, in many large sparse problems, discrete Newton ....

Coleman, T. F. ; Garbow, B. S. ; Mor'e, J. J. [1984]: Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software 11, pp. 363-378.

....column approach of this paper is a generalization of the Variable Isolation and the Element Isolation principles of Newsam and Ramsdell [3] In Section 5 we give a simple procedure to construct an expanded matrix from a given matrix. This enables us to use coloring routines implemented in [5] to find a partition of the segmented columns. In Section 6 we present computational results. Finally, in Section 7 we give some directions for further research. 2. Row partitioning. As has been pointed out earlier, CPR technique is based on grouping together of columns such that each group can ....

....zeros only. One consequence of Theorem 5.1 and Lemma 4.5 is that if Pi has m blocks then the resulting graph G(A Pi ) is isomorphic to GEI (A) Thus the expanded matrix procedure of this section provides an easy implementation of the EI principle. This helps us to use existing software [5] to compute the Jacobian matrix. 6. Computational Results. In this section we report the results from computational experiments for the segmented column approach. We test exact and heuristic coloring on G(A) G Pi (A) and G(A Pi ) 6.1. Test Suit. The test suit consists of matrices from the ....

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T. F. Coleman, B. S. Garbow and J. J. Mor'e, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 10 (1984), pp. 329-345.

....the product y T F 0 (x) when the vector y is initialized to jth coordinate vector. An excellent account of the recent developments in Automatic differentiation can be found in [5] JACOBIAN ESTIMATION 3 Much of the research on the efficient estimation of sparse Jacobian and Hessian matrices [1, 2, 4, 6, 7, 9, 10] uses divided differences to obtain estimates of the nonzeros. In this approach one forms groups of columns that are structurally orthogonal i.e. columns that do not have nonzero in the same row position. The estimates of the nonzeros in those columns are then obtained from a divided difference ....

....for finding complete direct cover that is described in this paper. The test matrices comprised of Harwell test collection. In the imple JACOBIAN ESTIMATION 11 mentation of the algorithm ties in the degree of vertices are broken arbitrarily. We also report test results obtained by applying dsm [4] on the test matrices (both A and A T ) The following statistics are generated. TABLE 1: Harwell Matrices (Static Information) Matrix n m nnz dnsm mxr mnr mng mxc mnc mng 0 abb313 176 313 1557 2.83 6 1 10 26 2 26 ash219 85 219 438 2.35 2 2 3 9 2 9 ash292 292 292 1250 1.47 8 1 8 10 1 10 ....

Coleman, T. F., Garbow, B. and Mor'e, J. J.(1984) , Software for estimating sparse Jacobian matrices, ACM Trans. Mathematical Software, 10 , pp. 329--347.

....is saved in g. Two schemes are tried (first fit and first fit after reverse column minimum degree ordering [19] and the more e#cient grouping is adopted. This may not result in an optimal grouping because finding the smallest packing is an NP complete problem equivalent to K coloring a graph [9]. The modified Rosenbrock code requires the partial derivative dFdt every time it requires dFdy. On reaching t, the step size h provides a measure of scale for the approximation of dFdt by a forward di#erence. The computation is so simple that it is done in the solver itself. 8. Examples. The ....

T. F. COLEMAN,B.S.GARBOW, AND J. J. MORE, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 11 (1984), pp. 329--345.

....techniques can be used to arrive at S and p. These algorithms produce a partitioning of the columns of the Jacobian into p structurally orthogonal groups by coloring the column intersection graph associated with the Jacobian. In our experiments we employ the graph coloring software described in [15] to obtain S, and then compute C(x) by initializing the ADIFOR generated derivative code of f(x) with the seed matrix set to S. From the point of view of computational complexity, the consequence of the compressed Jacobian approach is clear. Specifically, the memory and runtime requirements of ....

Thomas F. Coleman, Burton S. Garbow, and Jorge J. Mor'e. Software for estimating sparse Jacobian matrices. ACM Transactions on Mathematical Software, 10(3):329-- 345, 1984.

....for every assignment statement in the original function. Thus, if we compute a Jacobian J with n columns by setting g p = n, its computation will require roughly n times as many operations as the original function evaluation, independent of whether J is dense or sparse. However, it is well known [5,8] that the number of function evaluations that are required to compute an approximation to the Jacobian by finite differences can be much less than n if J is sparse. Fortunately, the same idea can be applied to greatly reduce the running time of ADIFOR generated derivative code as well. The idea is ....

Thomas F. Coleman, Burton S. Garbow, and Jorge J. Mor'e. Software for estimating sparse Jacobian matrices. ACM Transactions on Mathematical Software, 10(3):329--345, 1984.

.... seed matrix S is initialized by the user. So if S is the identity, ADIFOR computes the full Jacobian, and if S is just a vector, ADIFOR computes the product of the Jacobian by a vector. Compressed versions of sparse Jacobians can be computed by exploiting the same graph coloring techniques [12, 11] that are used for divided difference approximations of sparse Jacobians. The idea is best understood with an example. Assume that we have a function F = 0 B B B B f 1 f 2 f 3 f 4 f 5 1 C C C C A : x 2 R 4 7 y 2 R 5 whose Jacobian J has the following structure (symbols denote ....

T. F. Coleman, B. S. Garbow, and J. J. Mor'e. Software for estimating sparse Jacobian matrices. ACM Trans. Math. Software, 10:329 -- 345, 1984.

....function w T F , evaluated at x. TV=tensprod(fun,x,V,w,Extra) You can provide a full matrix, Extra, to be used by your (target) function fun . 24 9 References Here are some useful related references: ffl Automatic differentiation [10, 9] ffl Single sided determination of sparse Jacobians [5, 2] ffl Determination of sparse Hessian matrices [1, 4, 3] ffl Bi coloring [8] ffl Structured Jacobians, Hessians [7, 6] ffl ADOL C [11] ....

T. F. Coleman, B. S. Garbow, and J. J. Mor'e, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 10 (1984), pp. 329--345.

....orthogonal columns, we determine the Jacobian matrix by computing the compressed Jacobian matrix f E 0 (x)V , where V 2 R n Thetap . There is a column of V for each group, and v i;j 6= 0 only if the i th column of f E 0 (x) is in the j th group. Software for this partitioning problem [11] defines the groups with an array ngrp that sets the group for each column. The extended Jacobian can be determined from the compressed Jacobian matrix f E 0 (x)V by noting that if column j is in group k, then he i ; f E 0 (x)V e k i = v i;j i;j f E (x) Thus i;j f E (x) can be ....

T. F. Coleman, B. S. Garbow, and J. J. Mor' e, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 10 (1984), pp. 329--345.

.... mode ; given an arbitrary m by t W matrix W , the product W T J can be calculated using automatic differentiation in the reverse mode , e.g. Griewank1990a] Griewank1993a] The motivation for the bi coloring approach stems from the sparse finite differencing literature [Coleman1986a] [Coleman1984a], Coleman1985a] Coleman1984b] Coleman1984c] Curtis1974a] where graph coloring is used to partition the columns of a sparse Jacobian matrix J and subsequently define a matrix V such that J can be determined from the product JV . However, matrix V is not guaranteed to be thin, even if J has ....

T. F. Coleman, B. S. Garbow, and J. J. Mor' e, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 10 (1984), pp. 329--345.

....matrix of a mapping F : n m , is in the following terms : given a sparse m by n matrix J , obtain vectors d 1 ; d 2 ; d p such that the products Jd 1 ; Jd 2 ; Jd p determine J uniquely. This approach is called the one sided column approach for computing a sparse Jacobian [8, 4, 15]. Finite differences can be used to approximate the products Jd; automatic differentiation in the forward mode can be used to compute the products Jd exactly. The alternative row approach can be phrased: obtain vectors d 1 ; d 2 ; d p such that the products J T d 1 ; J T d 2 ; ....

....usually a few well chosen directions d 1 ; d 2 ; d p are needed to compute all the nonzeros of r 2 f(x) using the products r 2 f(x)d 1 ; r 2 f(x)d 2 ; r 2 f(x)d p . The algorithms that we have implemented are based on the work of Powell and Toint [21] and Coleman and Mor e [3, 7, 4]. These authors consider direct and indirect(substitution) methods; indirect methods usually require fewer function (or gradient) evaluations while direct methods produce more accurate approximations to the Hessian matrix. For a complete review on this subject, refer to Coleman and Cai [3] In ....

[Article contains additional citation context not shown here]

T. F. Coleman, B. S. Garbow, and J. J. Mor'e, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 10 (1984), pp. 329--345.

.... automatic differentiation in the forward mode ; given an arbitrary m by t W matrix W , the product W T J can be calculated using automatic differentiation in the reverse mode , e.g. 11, 12] The motivation for the bi coloring approach stems from the sparse finite differencing literature [4, 5, 6, 7, 8, 10] where graph coloring is used to partition the columns of a sparse Jacobian matrix J and subsequently define a matrix V such that J can be determined from the product JV . However, matrix V is not guaranteed to be thin, even if J has considerable sparsity : consider a sparse matrix J with a single ....

T. F. Coleman, B. S. Garbow, and J. J. Mor'e, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 10 (1984), pp. 329--345.

....for ADMIT Matlab help for all ADMIT functions is available. You can also get a table of contents of all ADMIT functions, by typing help ADMIT. 12 Other documents Here are some useful related references: ffl Automatic differentiation [10, 9] ffl Single sided determination of sparse Jacobians [5, 2] ffl Determination of sparse Hessian matrices [1, 4, 3] ffl Bi coloring [8] ffl Structured Jacobians, Hessians [7, 6] ffl ADOL C [11] 28 ....

T. F. Coleman, B. S. Garbow, and J. J. Mor'e, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 10 (1984), pp. 329--345.

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T. F. Coleman, B. S. Garbow and J. J. Mor'e [1984], "Software for estimating sparse Jacobian matrices", ACM Trans. Math. Software 11, pp. 363-378.

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T. F. Coleman, B. S. Garbow, and J. J. More. Software for estimating sparse Jacobian matrices. ACM Trans. Math. Software, 10(3):329--345, 1984.

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Thomas F. Coleman, Burton S. Garbow, and Jorge J. Mor'e. Software for estimating sparse Jacobian matrices. ACM Transactions on Mathematical Software, 10(3):329--345, 1984.

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T. F. Coleman, B. S. Garbow, and J. J. More. Software for estimating sparse Jacobian matrices. ACM Trans. Math. Software, 10(3):329--345, 1984.

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T. F. Coleman, B. Garbow, and J. J. More. Software for estimating sparse Jacobian matrices. ACM Trans. Mathematical Software, 10:329-- 347, 1984.

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T. F. COffMAN, B. S. GARBOW, AND J. J. MOR, Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software, 10 (1984), pp. 329-345.

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T. F. Coleman, B. S. Garbow and J. J. Mor'e [1984], "Software for estimating sparse Jacobian matrices", ACM Trans. Math. Software 11, pp. 363-378.

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Coleman,T. F.; Garbow, B. S.; Mor'e,J. J. [1984]: Software for estimating sparse Jacobian matrices, ACM Trans. Math. Software 11, pp. 363-378.

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