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This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Matrices. --- Numerical analysis. --- Mathematical analysis --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Matrices --- Numerical analysis --- Algorithm. --- Analysis of algorithms. --- Analytic function. --- Asymptotic analysis. --- Basis (linear algebra). --- Basis function. --- Biconjugate gradient method. --- Bidiagonal matrix. --- Bilinear form. --- Calculation. --- Characteristic polynomial. --- Chebyshev polynomials. --- Coefficient. --- Complex number. --- Computation. --- Condition number. --- Conjugate gradient method. --- Conjugate transpose. --- Cross-validation (statistics). --- Curve fitting. --- Degeneracy (mathematics). --- Determinant. --- Diagonal matrix. --- Dimension (vector space). --- Eigenvalues and eigenvectors. --- Equation. --- Estimation. --- Estimator. --- Exponential function. --- Factorization. --- Function (mathematics). --- Function of a real variable. --- Functional analysis. --- Gaussian quadrature. --- Hankel matrix. --- Hermite interpolation. --- Hessenberg matrix. --- Hilbert matrix. --- Holomorphic function. --- Identity matrix. --- Interlacing (bitmaps). --- Inverse iteration. --- Inverse problem. --- Invertible matrix. --- Iteration. --- Iterative method. --- Jacobi matrix. --- Krylov subspace. --- Laguerre polynomials. --- Lanczos algorithm. --- Linear differential equation. --- Linear regression. --- Linear subspace. --- Logarithm. --- Machine epsilon. --- Matrix function. --- Matrix polynomial. --- Maxima and minima. --- Mean value theorem. --- Meromorphic function. --- Moment (mathematics). --- Moment matrix. --- Moment problem. --- Monic polynomial. --- Monomial. --- Monotonic function. --- Newton's method. --- Numerical integration. --- Numerical linear algebra. --- Orthogonal basis. --- Orthogonal matrix. --- Orthogonal polynomials. --- Orthogonal transformation. --- Orthogonality. --- Orthogonalization. --- Orthonormal basis. --- Partial fraction decomposition. --- Polynomial. --- Preconditioner. --- QR algorithm. --- QR decomposition. --- Quadratic form. --- Rate of convergence. --- Recurrence relation. --- Regularization (mathematics). --- Rotation matrix. --- Singular value. --- Square (algebra). --- Summation. --- Symmetric matrix. --- Theorem. --- Tikhonov regularization. --- Trace (linear algebra). --- Triangular matrix. --- Tridiagonal matrix. --- Upper and lower bounds. --- Variable (mathematics). --- Vector space. --- Weight function.

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This volume is a collection of investigations involving the theory and applications of the various tools and techniques of mathematical analysis and analytic number theory, which are remarkably widespread in many diverse areas of the mathematical, biological, physical, chemical, engineering, and statistical sciences. It contains invited and welcome original as well as review-cum-expository research articles dealing with recent and new developments on the topics of mathematical analysis and analytic number theory as well as their multidisciplinary applications.

Research & information: general --- Mathematics & science --- subordination --- functions with positive real part --- reciprocals --- univalent functions --- starlikeness --- convexity --- close-to-convexity --- hyper-Bessel functions --- Hardy space --- distribution --- fractional Laplacian --- Riesz fractional derivative --- delta sequence --- convolution --- subordinations --- starlike functions --- convex functions --- close-to-convex functions --- cardioid domain --- Hankel determinant --- m-fold symmetric functions --- harmonic univalent functions --- with symmetric conjecture point --- integral expressions --- coefficient estimates --- distortion --- umbral methods --- harmonic numbers --- special functions --- integral representations --- laplace and other integral transforms --- analytic functions --- quasi-Hadamard --- differential operator --- closure property --- riemann zeta function --- asymptotics --- exponential sums --- multivalent functions --- q-Ruschweyh differential operator --- q-starlike functions --- circular domain --- q-Bernardi integral operator --- Bessel functions --- Appell–Bessel functions --- generating functions --- Chebyshev polynomials --- Euler sums --- Catalan’s constant --- Trigamma function --- integral representation --- closed form --- ArcTan and ArcTanh functions --- partial fractions --- Lambert series --- cotangent sum --- modular transformation --- Dedekind sum --- lemniscate of Bernoulli Hankel determinant --- determinant --- inverse --- Mersenne number --- periodic tridiagonal Toeplitz matrix --- Sherman-Morrison-Woodbury formula --- Fibonacci number --- Lucas number --- Toeplitz matrix --- Hankel matrix --- univalent function --- second Hankel determinant --- bi-close-to-convex functions --- gamma function and its extension --- Pochhammer symbol and its extensions --- hypergeometric function and its extensions --- τ-Gauss hypergeometric function and its extensions --- τ-Kummer hypergeometric function --- Fox-Wright function --- p-valent analytic function --- Hadamard product --- q-integral operator --- generalized Lupaş operators --- q analogue --- Korovkin’s type theorem --- convergence theorems --- Voronovskaya type theorem --- starlike function --- subordinate --- Janowski functions --- conic domain --- q-convex functions --- q-close-to-convex functions --- theta-function identities --- multivariable R-functions --- Jacobi’s triple-product identity --- Ramanujan’s theta functions --- q-product identities --- Euler’s pentagonal number theorem --- Rogers-Ramanujan continued fraction --- Rogers-Ramanujan identities --- combinatorial partition-theoretic identities --- Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities --- Schur’s second partition theorem --- subordination --- functions with positive real part --- reciprocals --- univalent functions --- starlikeness --- convexity --- close-to-convexity --- hyper-Bessel functions --- Hardy space --- distribution --- fractional Laplacian --- Riesz fractional derivative --- delta sequence --- convolution --- subordinations --- starlike functions --- convex functions --- close-to-convex functions --- cardioid domain --- Hankel determinant --- m-fold symmetric functions --- harmonic univalent functions --- with symmetric conjecture point --- integral expressions --- coefficient estimates --- distortion --- umbral methods --- harmonic numbers --- special functions --- integral representations --- laplace and other integral transforms --- analytic functions --- quasi-Hadamard --- differential operator --- closure property --- riemann zeta function --- asymptotics --- exponential sums --- multivalent functions --- q-Ruschweyh differential operator --- q-starlike functions --- circular domain --- q-Bernardi integral operator --- Bessel functions --- Appell–Bessel functions --- generating functions --- Chebyshev polynomials --- Euler sums --- Catalan’s constant --- Trigamma function --- integral representation --- closed form --- ArcTan and ArcTanh functions --- partial fractions --- Lambert series --- cotangent sum --- modular transformation --- Dedekind sum --- lemniscate of Bernoulli Hankel determinant --- determinant --- inverse --- Mersenne number --- periodic tridiagonal Toeplitz matrix --- Sherman-Morrison-Woodbury formula --- Fibonacci number --- Lucas number --- Toeplitz matrix --- Hankel matrix --- univalent function --- second Hankel determinant --- bi-close-to-convex functions --- gamma function and its extension --- Pochhammer symbol and its extensions --- hypergeometric function and its extensions --- τ-Gauss hypergeometric function and its extensions --- τ-Kummer hypergeometric function --- Fox-Wright function --- p-valent analytic function --- Hadamard product --- q-integral operator --- generalized Lupaş operators --- q analogue --- Korovkin’s type theorem --- convergence theorems --- Voronovskaya type theorem --- starlike function --- subordinate --- Janowski functions --- conic domain --- q-convex functions --- q-close-to-convex functions --- theta-function identities --- multivariable R-functions --- Jacobi’s triple-product identity --- Ramanujan’s theta functions --- q-product identities --- Euler’s pentagonal number theorem --- Rogers-Ramanujan continued fraction --- Rogers-Ramanujan identities --- combinatorial partition-theoretic identities --- Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities --- Schur’s second partition theorem

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This volume is a collection of investigations involving the theory and applications of the various tools and techniques of mathematical analysis and analytic number theory, which are remarkably widespread in many diverse areas of the mathematical, biological, physical, chemical, engineering, and statistical sciences. It contains invited and welcome original as well as review-cum-expository research articles dealing with recent and new developments on the topics of mathematical analysis and analytic number theory as well as their multidisciplinary applications.

subordination --- functions with positive real part --- reciprocals --- univalent functions --- starlikeness --- convexity --- close-to-convexity --- hyper-Bessel functions --- Hardy space --- distribution --- fractional Laplacian --- Riesz fractional derivative --- delta sequence --- convolution --- subordinations --- starlike functions --- convex functions --- close-to-convex functions --- cardioid domain --- Hankel determinant --- m-fold symmetric functions --- harmonic univalent functions --- with symmetric conjecture point --- integral expressions --- coefficient estimates --- distortion --- umbral methods --- harmonic numbers --- special functions --- integral representations --- laplace and other integral transforms --- analytic functions --- quasi-Hadamard --- differential operator --- closure property --- riemann zeta function --- asymptotics --- exponential sums --- multivalent functions --- q-Ruschweyh differential operator --- q-starlike functions --- circular domain --- q-Bernardi integral operator --- Bessel functions --- Appell–Bessel functions --- generating functions --- Chebyshev polynomials --- Euler sums --- Catalan’s constant --- Trigamma function --- integral representation --- closed form --- ArcTan and ArcTanh functions --- partial fractions --- Lambert series --- cotangent sum --- modular transformation --- Dedekind sum --- lemniscate of Bernoulli Hankel determinant --- determinant --- inverse --- Mersenne number --- periodic tridiagonal Toeplitz matrix --- Sherman-Morrison-Woodbury formula --- Fibonacci number --- Lucas number --- Toeplitz matrix --- Hankel matrix --- univalent function --- second Hankel determinant --- bi-close-to-convex functions --- gamma function and its extension --- Pochhammer symbol and its extensions --- hypergeometric function and its extensions --- τ-Gauss hypergeometric function and its extensions --- τ-Kummer hypergeometric function --- Fox-Wright function --- p-valent analytic function --- Hadamard product --- q-integral operator --- generalized Lupaş operators --- q analogue --- Korovkin’s type theorem --- convergence theorems --- Voronovskaya type theorem --- starlike function --- subordinate --- Janowski functions --- conic domain --- q-convex functions --- q-close-to-convex functions --- theta-function identities --- multivariable R-functions --- Jacobi’s triple-product identity --- Ramanujan’s theta functions --- q-product identities --- Euler’s pentagonal number theorem --- Rogers-Ramanujan continued fraction --- Rogers-Ramanujan identities --- combinatorial partition-theoretic identities --- Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities --- Schur’s second partition theorem