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QA297 .B954 2013
Numerical analysis
2011
Burden, Richard L.
Math & Statistics Library » QA297 .B84 2011
Numerical methods and optimization
2014
Butenko, Sergiy
Math & Statistics Library » QA297 .B845 2014
Condition
2013
Bürgisser, Peter, 1962
Math & Statistics Library » QA297 .B954 2013
Principles of computation
1965
Calingaert, Peter
Math & Statistics Library » QA297 .C3
Lectures on numerical analysis and ...
1965
Capriz, G. (Giuseppe)
SAL3 (offcampus storage) » QA297 .C325
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Condition : the geometry of numerical algorithms / by Peter Bürgisser, Felipe Cucker.
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QA297 .B954 2013
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Author/Creator:
Bürgisser, Peter, 1962
author.
Language:
English.
Publication date:
2013
Publication:
Berlin : Springer, 2013.
Format:
Book
xxxi, 554 pages : illustrations ; 25 cm.
Contents:
Condition in linear algebra (adagio)
Normwise condition of linear equation solving
Vector and matrix norms
Turing's condition number
Condition and distance to illposedness
An alternative characterization of condition
The singular value decomposition
Least squares and the MoorePenrose inverse
Probabilistic analysis
A crash course on integration
A crash course on probability : I
Basic facts
Gaussian distributions
The X² Distribution
Uniform distributions on spheres
Expectations of nonnegative random variables
Caps and tubes in spheres
Average and smoothed analyses
Probabilistic analysis of Cwi (A, x)
Probabilistic analysis of Krs (A)
Preconditioning
Average analysis
Uniform smoothed analysis
Additional considerations
Probabilistic analysis for other norms
Probabilistic analysis for Gaussian distributions
Error analysis of triangular linear systems
Random triangular matrices are illconditioned
Backward analysis of triangular linear systems
Componentwise condition of random sparse matrices
Componentwise condition numbers
Determinant computation
Matrix inversion
Solving linear equations
Error bounds for triangular linear systems
Additional considerations
On norms and mixed condition numbers
On the underlying probability measure
Probabilistic analysis of rectangular matrices
A crash course on probability : II
Large deviations
Random Gaussian matrices
A bound on the expected spectral norm
Tail bounds for k(A)
Tail bounds for (A)
Proof of theorem 4.16
Expectations : proof of theorem 4.2
Complex matrices
Condition numbers and iterative algorithms
The cost of computing : a primer in complexity
The method of steepest descent
The method of conjugate gradients
Conjugate gradient on random data
Intermezzo I : condition of structured data
Condition in linear optimization (Andante)
A condition number for polyhedral conic systems
Condition and continuity
Basic facts on convexity
Convex sets
Polyhedra
The polyhedral cone feasibility problem
The GCC condition number and distance to illposedness
The GCC condition number and spherical caps
The GCC condition number and images of balls
The GCC condition number and wellconditioned solutions
Condition of solutions and condition numbers
The perceptron algorithm for feasible cones
The ellipsoid method
A few facts about ellipsoids
The ellipsoid method
Polyhedral conic systems with integer coefficients
Linear programs and their solution sets
Linear programs and duality
The geometry of solution sets
The combinatorics of solution sets
Illposedness and degeneracy
Degeneracy
A brief discussion on illposedness
Interiorpoint methods
Primaldual interiorpoint methods : basic ideas
Existence and uniqueness of the central path
Analysis of IPM for linear programming
Conditionbased analysis of IPM for PCFP
Reformulation
Algorithmic solution
Analysis
Finite precision for decision and counting problems
The linear programming feasibility problem
A condition number for polyhedral feasibility
Deciding feasibility of primaldual pairs
Condition and linear programming optimization
The condition number K (d)
K(d) and optimal solutions
Computing the optimal basis
An interiorpoint algorithm
A reduction to polyhedral feasibility problems
Optimizers and optimal bases : the condition viewpoint
Approximating the optimal value
Average analysis of the RCC condition number
Proof of theorem 12.1
The group ... and its action
Probabilities
Probabilistic analyses of the GCC condition number
The probability of primal and dual feasibility
Spherical convexity
A bound on the volume of tubes
Two essential reductions
A crash course on probability : III
Average analysis
Smoothed analysis
Intermezzo II : the condition of the condition
Condition in polynomial equation solving (allegro con brio)
A geometric framework for condition numbers
Condition numbers revisited
Complex zeros of univariate polynomials
A geometric framework
Linear equation solving
Complex projective space
Projective space as a complex manifold
Distances in projective space
Condition measures on manifolds
Eigenvalues and eigenvectors
Computation of the Kernel
Homotopy continuation and Newton's method
Homotopy methods
Newton's method
Homogeneous polynomial systems
A unitarily invariant inner product
A unitarily invariant condition number
Orthogonal decompositions of Hd
A condition number theorem
Bézout's theorem
A projective Newton's method
A higher derivative estimate
A lipschitz estimate for the condition number
Smale's 17th problem : I
The adaptive linear homotopy for Hd
Interlude : randomization
Randomized algorithms
A Las Vegas homotopy method
A crash course on probability : IV
Normal Jacobians of projections
The standard distribution on the solution variety
Beltránpardo randomization
Analysis of algorithm LV
Average analysis of unorm, uav, and umax
Smale's 17th problem : II
The main technical result
Outline of the proof
Normal Jacobians of linearizations
Induced probability distributions
Smoothed analysis of LV
Conditionbased analysis of LV
A nearsolution to smale's 17th problem
A deterministic homotopy continuation
An elimination procedure for zerofinding
Some inequalities of combinatorial numbers
Real polynomial systems
Homogeneous systems with real coefficients
On the condition for real zerocounting
Smale's atheory
An algorithm for real zerocounting
Grids and graphs
Proof of theorem 19.1
On the average number of real zeros
Feasibility of underdetermined and semialgebraic systems
Probabilistic analysis of conic condition numbers : I. the complex case
The basic idea
Volume of tubes around linear subspaces
Volume of algebraic varieties
A crash course on probability : V
Proof of theorem 20.1
Applications
Linear equationsolving
Eigenvalue computations
Complex polynomial systems
Probabilistic analysis of conic condition numbers : II. the real case
On the volume of tubes
Curvature Integrals
Weyl's tube formula
A crash course on probability : VI
Bounding integrals of curvature
Proof of theorem 21.1
The smooth case
The general case
Proof of theorem 21.1
An application
Tubes around convex sets
Integrals of curvature for boundaries of convex sets
Proof of theorem 13.18
Conic condition numbers and structured data
Smoothed analysis for adversarial distributions
Appendix
Big oh, little oh, and other comparisons
Differential geometry
Submanifolds of Rn
Abstract smooth manifolds
Integration on manifolds
Sard's theorem and transversality
Riemannian metrics
Orthogonal and unitary groups
Curvature of hypersurfaces
Algebraic geometry
Varieties
Dimension and regular points
Elimination theory
Degree
Resultant and discriminant
Volumes of complex projective varieties
Integral geometry
Poincaré's formula
The principal kinematic formula
Notes
Coda : open problems
Probabilistic analysis of growth factors
Eigenvalue problem
Smale's 9th problem
Smoothed analysis of RCC condition number
Improved average analysis of Grassmann condition
Smoothed analysis of Grassmann condition
Robustness of condition numbers
Average complexity of IPMs for linear programming
Smale's 17th problem
The shubsmale starting system
Equivariant Morse function
Good starting pairs in one variable
Approximating condition geodesies
Selfconvexity of unorm in higher degrees
Structured systems of polynomial equations
Systems with singularities
Conic condition numbers of real problems with high codimension of illposedness
Feasibility of real polynomial systems
Bibliography
Notation
 concepts
 and the people who crafted them.
Contributor:
Cucker, Felipe, 1958
author.
Series:
Grundlehren der mathematischen Wissenschaften ;
349.
Subjects:
Numerical analysis.
Algorithms.
ISBN:
9783642388958
3642388957
Catkey: 10281923
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