M. Iri, Simultaneous Computation of Functions, Partial Derivatives and Estimates of Rounding Errors -- Complexity and Practicality, Japan Journal of Applied Mathematics, Vol. 1, No. 2, pp. 223-252, 1984  Home/Search   Document Not in Database   Summary   Related Articles   Check

....da[1] 2 x y; da[2] x2; a : x2 y; t1 : array(1 . 2,1 . 2) t1[1,1] 6 a y da[1]2 3 a2 y dda[1,1] t1[1,2] 6 a y da[2] da[1] 3 a2 y dda[1,2] 3 a2 da[1] t1[2,1] t1[1,2] t1[2,2] 6 a y da[2]2 6 a2 da[2] t1 end eval(hessf(x,y) 4 4 5 3 ] 30 x y 24 x y ] [ 5 3 6 2 ] [ 24 x y 12 x y ] 5 Conclusion It is natural to consider doing Algorithmic Differentiation in a computer algebra system because the system already knows how to differentiate and simplify formulae. In designing and implementing the GRADIENT routine, we have found this to be a great advantage. ....

M. Iri, Simultaneous Computation of Functions, Partial Derivatives and Estimates of Rounding Errors -- Complexity and Practicality, Japan Journal of Applied Mathematics, Vol. 1, No. 2, pp. 223-252, 1984

....2 ### 2 ) 16) where the derivatives of # with respect to u i are obtained from (12) and #a, b# denotes the inner product of vectors a and b. This estimate can be used instead of a laborious interval analysis. Theoretical and practical aspects of error estimation were investigated by Iri [13]. The formulas (15) 16) and similar expressions below have been used often in debugging various programs for computing vectors p i . In problems where the di#erentiation of functions F is di#cult, the codes for calculating the adjoint variables according to (11) are not correct at times, and we ....

....analyze various algorithms for automatic di#erentiation from the point of view of the algebraic complexity of the computation, i.e. the total number of arithmetic operations required to compute a function and its partial derivatives. For results in this field, we refer the interested readers to [2, 9, 10, 11, 12, 13, 14, 15] and the relevant references cited therein. Let T 0 denote the total time required to calculate the value of the underlying function f(u) Let T g denote the additional time required for computing all partial derivatives #f(u) #u i , 1 # i # r. Theorem 3.1 Suppose that 1. f(u) is an ....

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M. Iri, Simultaneous computation of functions, partial derivatives and estimates of rounding errors --- Complexity and practicality, Japan Journal of Applied Mathematics, 1, 223-252 (1984).

....The model is: f:A; B; Cg. fBg has 2 models since G(S) VB = 2. The models are: fA; B; Cg and f:A; B; Cg. There is one missing link now: How do we compute the partial derivatives of a counting graph with respect to each of its variables This actually turns out to be easy due to results in [14, 16] which show how to evaluate and simultaneously compute all partial derivatives of a function by simply traversing its computation graph in linear time. Although [16] casts such computation in terms of summing weights of paths in such a graph, we present a more direct implementation here. In ....

Masao. Simultaneous computation of functions, partial derivatives and estimates of rounding error. Japan J. Appl. Math., 1:223-252, 1984.

....kffik 2 ) 16) where the derivatives of Omega with respect to u i are obtained from (12) and ha; bi denotes the inner product of vectors a and b. This estimate can be used instead of a laborious interval analysis. Theoretical and practical aspects of error estimation were investigated by Iri [13]. The formulas (15) 16) and similar expressions below have been used often in debugging various programs for computing vectors p i . In problems where the differentiation of functions F is difficult, the codes for calculating the adjoint variables according to (11) are not correct at times, ....

....analyze various algorithms for automatic differentiation from the point of view of the algebraic complexity of the computation, i.e. the total number of arithmetic operations required to compute a function and its partial derivatives. For results in this field, we refer the interested readers to [2, 9, 10, 11, 12, 13, 14, 15] and the relevant references cited therein. Let T 0 denote the total time required to calculate the value of the underlying function f(u) Let T g denote the additional time required for computing all partial derivatives f(u) u i , 1 i r. Theorem 3.1 Suppose that 1. f(u) is an elementary ....

[Article contains additional citation context not shown here]

M. Iri, Simultaneous computation of functions, partial derivatives and estimates of rounding errors --- Complexity and practicality, Japan Journal of Applied Mathematics, 1, 223-252 (1984).

....closely related to the fast automatic differentiation Sponsored by FAPESP (Grants 95 2452 6 and 97 12033 6) y Sponsored by the Russian Foundation for Basic Research (Grants 96 15 96124 and 98 01 00517) and FAPESP (Grant 96 6631 5) 1 e.g. BIRGIN AND Y. G. EVTUSHENKO (FAD) techniques (see [25, 26, 21, 27, 22]) In [13] using generalized FAD expressions, the exact gradient of the objective function of a general multistep process was derived in a very simple canonical form. One of the aims of this paper is to show the application of these canonical formulas to optimal control processes being integrated ....

M. Iri, Simultaneous computation of functions, partial derivatives and estimates of rounding errors - Complexity and practicality, Japan Journal of Applied Mathematics 1 (1984), pp. 223--252.

.... technique (1970, in Finnish) Since then the reverse mode has been rediscovered and restated many times, e.g. 1971 by Ostrowski, Wolin, and Borisow [12] 1972 by Tienari [16] 1974 by Werbos [17] 1980 by Miller and Wrathall [11] and by Speelpenning [15] 1983 by Baur and Strassen [1] 1984 by Iri [7], by Kim, Nesterov, and Cherkasskii [8] and by Sawyer [14] In the present paper we investigate the complexity of algorithms for computing the derivative of a rational function. In particular we deal with the forward mode and the reverse mode. We give bounds on the amount of work within the ....

Iri, M.: Simultaneous computation of functions, partial derivatives and estimates of rounding errors, complexity and practicality. Japan J. Appl. Math. 1, 1984, 223--252.

....be the rounding error generated when computing the value of node i that is, the difference between the value computed using limitedprecision and in nite precision. If j i j j val(i) j for a machine speci c , then the rounding error in computing F(e; F(e; can be bound as follows (see [17] for details) F(e; X non leaf i j pd(i)val(i) j : Note that bounding the rounding error can be done simultaneously during the passage of pd messages, and requires no extra space. Our nal remark is on the computation of second partial derivatives. The partial derivative pd(i) is by ....

....and can be di erentiated again using the same approach. In particular, if the compilation F has O(n) variables, then its second partial derivatives can be computed simultaneously in O(n j F j) time and space, where j F j is the size of F . Again, this follows from a more general result in [17]. Therefore, if the compilation F of a Bayesian network of size n is induced using an elimination order of width w, as suggested earlier, then the value of polynomial F ; all its rst partial derivatives; and all its second partial derivatives can be computed simultaneously in O(n 2 exp(w) ....

Masao. Simultaneous computation of functions, partial derivatives and estimates of rounding error. Japan J. Appl. Math., 1:223-252, 1984.

.... an ill posed problem of numerical di erentiation by applying well known methods of regularization; see, e.g. 13] Moreover, in cases where the gradient can be calculated in the same number of operations as the function itself, computational e ort can be minimized by using methods described in [12]. To estimate the round o error (3) T , we apply the well known estimates [19] of the type m X k=1 a k l m X k=1 a k 1:06 2 l max 1 k m ja k j (m 2) 2 2 ; j(a k b k ) l a k b k j 2 l ja k b k j; 2 l m 0:1 to scheme (16) As a result, we obtain the ....

M. Iri. Simultaneous computation of functions, partial derivatives and estimates of rounding errors | complexity and practicality, Japan J. of Applied Mathematics, 1(1984), N 2, 223-252.

....the sixties it was common to justify the existence of most quasiNewton methods saying that the task of computing derivatives is prone to human errors. However, automatic differentiation techniques have been developed in the last 20 years that, in practice, eliminates the possibility of error. See [31, 45, 50, 87, 88, 89] and many others. Moreover, in most cases, the computation of derivatives using automatic differentiation is not expensive. This implies that, in modern practice, the most interesting quasiNewton methods are those in which the Jacobian approximations are defined in such a way that much linear ....

Iri, M. [1984]: Simultaneous computations of functions, partial derivatives and estimates of rounding errors. Complexity and Practicality , Japan Journal of Applied Mathematics 1, pp. 223 - 252.

....the rounding error generated when computing the value of node i that is, the di erence between the value computed using limited precision and in nite precision. If j i j j val(i) j for a machine speci c , then the rounding error in computing F(e; F(e; can be bound as follows (see [20] for details) F(e; X non leaf i j pd(i)val(i) j : Note that bounding the rounding error can be done simultaneously during the passage of pd messages, and requires no extra space. Our nal remark is on the computation of second partial derivatives. The partial derivative pd(i) is by ....

....and can be di erentiated again using the same approach. In particular, if the compilation F has O(n) variables, then its second partial derivatives can be computed simultaneously in O(n j F j) time and space, where j F j is the size of F . Again, this follows from a more general result in [20]. Therefore, if the compilation F of a Bayesian network of size n is induced using an elimination order of induced width w, as suggested earlier, then the value of polynomial F ; all its rst partial derivatives; and all its second partial derivatives can be computed simultaneously in O(n 2 ....

Masao. Simultaneous computation of functions, partial derivatives and estimates of rounding error. Japan J. Appl. Math., 1:223-252, 1984.

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Iri, M. [1984]: Simultaneous computations of functions, partial derivatives and estimates of rounding errors. Complexity and Practicality , Japan Journal of Applied Mathematics 1, pp. 223 - 252.

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