We extend Smale’s concept of approximate zeros of an analytic function on a Banach space to two c... more We extend Smale’s concept of approximate zeros of an analytic function on a Banach space to two computational models that account for errors in the computation: first, the weak model where the computations are done with a fixed precision; and second, the strong model where the computations are done with varying precision. For both models, we develop a notion of robust approximate zero and derive a corresponding robust point estimate. A useful specialization of an analytic function on a Banach space is a system of integer polynomials. Given such a zero-dimensional system, we bound the complexity of computing an absolute approximation to a root of the system using the strong model variant of Newton’s method initiated from a robust approximate zero. The bound is expressed in terms of the condition number of the system and is a generalization of a well-known bound of Brent to higher dimensions.
This paper presents two results on the complexity of root isolation via Sturm sequences. Both res... more This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments. For a square-free polynomial A (X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most O(dL + dlgd) Sturm probes. This extends Davenport’s result for the case of isolating all real roots. We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d 3L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.
Smale’s notion of an approximate zero of an analytic function f: C →C is extended to take into ac... more Smale’s notion of an approximate zero of an analytic function f: C →C is extended to take into account the errors incurred in the evaluation of the Newton operator. Call this stronger notion a robust approximate zero.. We develop a corresponding robust point estimate for such zeros: we prove that if z 0 ∈ C satisfies α(f,z 0)<0.02 then z 0 is a robust approximate zero, with the associated zero z * lying in the closed disc \({\overline B}(z_{0},\frac{0.07}{f,z_{0}}\) . Here α(f,z), γ(f,z) are standard functions in point estimates. Suppose f(z) is an L-bit integer square-free polynomial of degree d. Using our new algorithm, we can compute an n-bit absolute approximation of z * ∈ IR starting from a bigfloat z 0, in time O[dM(n + d 2(L + lg d)lg(n + L))], where M(n) is the complexity of multiplying n-bit integers.
We extend Smale’s concept of approximate zeros of an analytic function on a Banach space to two c... more We extend Smale’s concept of approximate zeros of an analytic function on a Banach space to two computational models that account for errors in the computation: first, the weak model where the computations are done with a fixed precision; and second, the strong model where the computations are done with varying precision. For both models, we develop a notion of robust approximate zero and derive a corresponding robust point estimate. A useful specialization of an analytic function on a Banach space is a system of integer polynomials. Given such a zero-dimensional system, we bound the complexity of computing an absolute approximation to a root of the system using the strong model variant of Newton’s method initiated from a robust approximate zero. The bound is expressed in terms of the condition number of the system and is a generalization of a well-known bound of Brent to higher dimensions.
This paper presents two results on the complexity of root isolation via Sturm sequences. Both res... more This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments. For a square-free polynomial A (X) of degree d with L-bit integer coefficients, we use an amortization argument to show that all the roots, real or complex, can be isolated using at most O(dL + dlgd) Sturm probes. This extends Davenport’s result for the case of isolating all real roots. We also show that a relatively straightforward algorithm, based on the classical subresultant PQS, allows us to evaluate the Sturm sequence of A(X) at rational Õ(dL)-bit values in time Õ(d 3L); here the Õ-notation means we ignore logarithmic factors. Again, an amortization argument is used. We provide a family of examples to show that such amortization is necessary.
Smale’s notion of an approximate zero of an analytic function f: C →C is extended to take into ac... more Smale’s notion of an approximate zero of an analytic function f: C →C is extended to take into account the errors incurred in the evaluation of the Newton operator. Call this stronger notion a robust approximate zero.. We develop a corresponding robust point estimate for such zeros: we prove that if z 0 ∈ C satisfies α(f,z 0)<0.02 then z 0 is a robust approximate zero, with the associated zero z * lying in the closed disc \({\overline B}(z_{0},\frac{0.07}{f,z_{0}}\) . Here α(f,z), γ(f,z) are standard functions in point estimates. Suppose f(z) is an L-bit integer square-free polynomial of degree d. Using our new algorithm, we can compute an n-bit absolute approximation of z * ∈ IR starting from a bigfloat z 0, in time O[dM(n + d 2(L + lg d)lg(n + L))], where M(n) is the complexity of multiplying n-bit integers.
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Papers by Vikram Sharma