Generalization of a Theorem on Eigenvalues of Symmetric Matrices
Rohn, Jiří
2019 - English
We prove that the product of a symmetric positive semide nite matrix and a symmetric matrix has all eigenvalues real.
Keywords:
symmetric matrix; positive semide nite matrix; real spectrum
Available in a digital repository NRGL
Generalization of a Theorem on Eigenvalues of Symmetric Matrices
We prove that the product of a symmetric positive semide nite matrix and a symmetric matrix has all eigenvalues real.
Hybrid Methods for Nonlinear Least Squares Problems
Lukšan, Ladislav; Matonoha, Ctirad; Vlček, Jan
2019 - English
This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function F(x) = (1=2)fT (x)f(x), where x ∈ Rn and f ∈ Rm, together with extensive computational tests and comparisons of the introduced methods. All hybrid methods are described in detail and their global convergence is proved in a unified way. Some proofs concerning trust region methods, which are difficult to find in the literature, are also added. In particular, the report contains an analysis of a new simple hybrid method with Jacobian corrections (Section 8) and an investigation of the simple hybrid method for sparse least squares problems proposed previously in [33] (Section 14).
Keywords:
numerical optimization; nonlinear least squares; trust region methods; hybrid methods; sparse problems; partially separable problems; numerical experiments
Available in a digital repository NRGL
Hybrid Methods for Nonlinear Least Squares Problems
This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function F(x) = (1=2)fT (x)f(x), where x ∈ Rn and f ∈ Rm, together with ...
Does a Singular Symmetric Interval Matrix Contain a Symmetric Singular Matrix?
Rohn, Jiří
2019 - English
We consider the conjecture formulated in the title concerning existence of a symmetric singular matrix in a singular symmetric interval matrix. We show by means of a counterexample that it is generally not valid, and we prove that it becomes true under an additional assumption of positive semide niteness of the midpoint matrix. The proof is constructive.
Keywords:
symmetric interval matrix; singularity; positive semide niteness
Available in a digital repository NRGL
Does a Singular Symmetric Interval Matrix Contain a Symmetric Singular Matrix?
We consider the conjecture formulated in the title concerning existence of a symmetric singular matrix in a singular symmetric interval matrix. We show by means of a counterexample that it is ...
Transforming hierarchical images to program expressions using deep networks
Křen, Tomáš
2018 - English
We present a technique describing how to effectively train a neural network given an image to produce a formal description of the given image. The basic motivation of the proposed technique is an intention to design a new tool for automatic program synthesis capable of transforming sensory data (in our case static image, but generally a phenotype) to a formal code expression (i.e. syntactic tree of a program), such that the code (from evolutionary perspective a genotype) evaluates to a value that is similar to the input data, ideally identical. Our approach is partially based on our technique for generating program expressions in the context of typed functional genetic programming. We present promising results evaluating a simple image description language achieved with a deep network combining convolution encoder of images and recurrent decoder for generating program expressions in the sequential prefix notation and propose possible future applications.
Keywords:
deep networks; automatic program synthesis; image processing
Available in a digital repository NRGL
Transforming hierarchical images to program expressions using deep networks
We present a technique describing how to effectively train a neural network given an image to produce a formal description of the given image. The basic motivation of the proposed technique is an ...
Sparse Test Problems for Nonlinear Least Squares
Lukšan, Ladislav; Matonoha, Ctirad; Vlček, Jan
2018 - English
This report contains a description of subroutines which can be used for testing large-scale optimization codes. These subroutines can easily be obtained from the web page http://www.cs.cas.cz/~luksan/test.html. Furthermore, all test problems contained in these subroutines are presented in the analytic form.
Keywords:
large-scale optimization; least squares; test problems
Available in a digital repository NRGL
Sparse Test Problems for Nonlinear Least Squares
This report contains a description of subroutines which can be used for testing large-scale optimization codes. These subroutines can easily be obtained from the web page ...
Problems for Nonlinear Least Squares and Nonlinear Equations
Lukšan, Ladislav; Matonoha, Ctirad; Vlček, Jan
2018 - English
This report contains a description of subroutines which can be used for testing large-scale optimization codes. These subroutines can easily be obtained from the web page http://www.cs.cas.cz/~luksan/test.html. Furthermore, all test problems contained in these subroutines are presented in the analytic form.
Keywords:
large-scale optimization; least squares; nonlinear equations,; test problems
Available in a digital repository NRGL
Problems for Nonlinear Least Squares and Nonlinear Equations
This report contains a description of subroutines which can be used for testing large-scale optimization codes. These subroutines can easily be obtained from the web page ...
A limited-memory optimization method using the infinitely many times repeated BNS update and conjugate directions
Vlček, Jan; Lukšan, Ladislav
2018 - English
Keywords:
Unconstrained minimization; variable metric methods; limited-memory methods; the repeated BFGS update; global convergence; numerical results
Available in digital repository of the ASCR
A limited-memory optimization method using the infinitely many times repeated BNS update and conjugate directions
Numerical solution of generalized minimax problems
Lukšan, Ladislav; Matonoha, Ctirad; Vlček, Jan
2018 - English
Keywords:
Numerical optimization; nonlinear approximation; nonsmooth optimization; generalized minimax problems; recursive quadratic programming methods; interior point methods; smoothing methods; algorithms; numerical experiments
Available in digital repository of the ASCR
Numerical solution of generalized minimax problems
Semigroup Structure of Sets of Solutions to Equation X^s = X^m
Porubský, Štefan
2017 - English
Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over multiplicative semigroups of factor rings of residually finite commutative rings and finally over multiplicative semigroups of factor rings of residually finite commutative principal ideal domains. The analysis is done through the use of the maximal subsemigroups and groups corresponding to an idempotent of the corresponding semigroup and in the case of residually finite PID’s employing the available analysis of the Euler-Fermat Theorem as given in [11]. In particular the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to x^n = x connected with the message space of RSA cryptosystems and Fermat pseudoprimes.
Keywords:
set of solutions; idempotent; maximal semigroup corresponding to an idempotent; maximal group corresponding to an idempotent; equation X^s = X^m; finite commutative ring with identity element; residually finite commutative principal ideal domains
Available on request at various institutes of the ASCR
Semigroup Structure of Sets of Solutions to Equation X^s = X^m
Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over ...
Idempotents, Group Membership and their Applications
Porubský, Štefan
2017 - English
S.Schwarz in his paper [165] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought [167] into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.
Keywords:
multiplicative semigroup; finite semigroups; power semigroups; idempotent elements; finite commutative rings; principal ideal domain; Euler-Fermat theorem; Wilson theorem; matrices over fields; maximal groups contained in a semigroup; periodic sequence; multiplicative semigroup of Zm; semigroup of circulant Boolean matrices
Available on request at various institutes of the ASCR
Idempotents, Group Membership and their Applications
S.Schwarz in his paper [165] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought [167] into play the maximal subsemigroups and thus he embodied the ...
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