Principles of Mathematics in Operations Research
5 Positive Definiteness Positive definite matrices are of both theoretical and computational impor- tance in a wide variety of a ...
72 5 Positive Definiteness Example 5.1.1 Let f(x,y) = x^2 + y^2. Find the extreme points of f(x,y): ox ay Since we have only one ...
5.1 Minima, Maxima, Saddle points 73 Theorem 5.1.3 The extreme values for f(x,y) can occur only at i. Boundary points of the dom ...
74 5 Positive Definiteness Remark 5.1.6 Let f : M™ M- R andx G Rn be the local minimum, Vf(x) = 0 and V^2 f(x) is positive defin ...
5.3 Semidefinite Matrices 75 cTAx •• [xk,0] A* * k * Xk 0 = x\AkXk > 0. If we apply (i) <=> (ii) for Ak (its eigen valu ...
76 5 Positive Definiteness Remark 5.3.3 (Indefinite matrices) Change of Variables: y = Cx. The quadratic form becomes yTCTACy. T ...
Web material Remark 5.4.6 Xj = minxSRn R(x) \j = maxl6r R(x) s.t. s.t. xTv\ = 0 xTvj+i = 0 xTVj-i = 0 xTvn = 0 Problems 5.1. Pro ...
78 5 Positive Definiteness http://eom.springer.de/N/nl30030.htm http://eom.springer.de/q/q076080.htm http://epubs.siam.org/sam-b ...
5.5 Web material 79 http://www.chass.utoronto.ca/~osborne/MathTutorial/QFF.HTM http://www.chass.utoronto.ca/"osborne/MathTutoria ...
6 Computational Aspects For square matrices, we can measure the sensitivity of the solution of the linear algebraic system Ax = ...
82 6 Computational Aspects n A/, — 2_.aivi where Vi <-> A,, 2_aj ~ ^' a> — 0;Vi. i=l If A), is along i>i, i.e. At, — ...
6.1 Solution of Ax = b^83 \\x\\ = -y> Ax = x 2 -xi = - = 1 ana 1 -1 =* IIArll = V2 Ml 5 x 10~^6. T/ie relative amplification ...
84 6 Computational Aspects Definition 6.1.7 The norm of A is the number defined \\A\\ = xnaxx^o |d!f Remark 6.1.8 ||.A|| bounds ...
6.1 Solution of Ax = b 85 Proposition 6.1.13 The norm of A is the square root of the largest eigen value of ATA. The vector that ...
86 6 Computational Aspects A~ 1_ 999 10 10989 100 10989 999 u 1010 1000 999 99 100 999 100 99 AA = \A~X -1 - J\max[(A-i)TA] = ...
Ai = ai I A:' Xl 6.2 Computation of eigen values 87 The vectors uk point more and more accurately towards the direction of xn an ...
88 6 Computational Aspects Remark 6.2.3 (QR Algorithm) Start with Ao. Factor it using the Gram- Schmidt process into QQRQ, then ...
6.2 Problems 89 The second stage is similar: x consists of the last n — 2 entries in the second column, z is the first unit coor ...
90 6 Computational Aspects Web material http://202.41.85.103/manuals/planetmath/entries/65/ MatrixConditionNumber/MatrixConditio ...
6.3 Web material http: //www. cs. unc. edu/~krishnas/eigen/node6 .html http://www.cs.ut.ee/"toomas_l/linalg/linl/nodel8.html htt ...
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