4. RESULTADOS Y DISCUSION
4.6 Evaluación sensorial de la carne
In this section we sketch briefly several instances in which the KM The- orem can be applied to obtain convergence of iterative algorithms. These methods will be considered in more detail later in the text.
10.10.1
The ART
The ART is an iterative method for solving Ax = b, a system of I
linear equations inJ unknowns. For eachi= 1, ..., I, we denote byPi the
orthogonal projection onto the hyperplane
Hi={x|(Ax)i =bi},
and let
T =PIPI−1· · ·P2P1.
Thefull-cycleART sequence is defined aszm=Tmz0, form= 1,2, .... The
operators Pi are averaged, therefore so is T. If the system has solutions,
thenT has fixed points, and, by the KM Theorem 10.3, the sequence{zm}
converges to a solution of Ax=b. Later, we shall prove that the limit is the solution closest toz0 in the 2-norm.
10.10.2
The CQ Algorithm
Let Abe a real I byJ matrix, and C and Qgiven closed convex sets in RJ and
RI, respectively. The split feasibility problem(SFP) is to find
xin C with Ax in Q. In [62] the CQ algorithm for solving the SFP was presented, for the real case. It has the iterative step
xk+1=PC(xk−γAT(I−PQ)Axk), (10.50)
where I is the identity operator and γ ∈ (0,2/ρ(ATA)), for ρ(ATA) the
spectral radius of the matrixATA, which is also its largest eigenvalue. The
has complex entries, and the setsC andQare inCJ andCI, respectively. The iterative step of the extended CQ algorithm is then
xk+1=PC(xk−γA†(I−PQ)Axk). (10.51)
The CQ algorithm converges to a solution of the SFP, for any starting vectorx0, whenever the SFP has solutions. When the SFP has no solutions,
the CQ algorithm converges to a minimizer of the function
f(x) =1
2||PQAx−Ax||
2 2
over the set C, provided such constrained minimizers exist [62, 63]. The CQ algorithm employs the relaxation parameterγin the interval (0,2/L), where L is the largest eigenvalue of the matrix ATA. Choosing the best
relaxation parameter in any algorithm is a nontrivial procedure. Generally speaking, we want to selectγ near to 1/L.
For the real case, the gradient of the functionf(x) is
∇f(x) =AT(I−PQ)Ax. (10.52)
It is shown in [63] that the operator ∇f(x) is ρ(ATA)-Lipschitz and therefore is ν-ism for ν = 1/ρ(ATA). It follows that the operator T =
PC(I−γ∇f) is averaged, whenever 0< γ <2/ρ(ATA). Consequently, the
CQ algorithm converges to a minimizer off(x) overxinC, whenever such minimizers exist.
WithA=I andC=Q, it follows that the gradient of the function
h(x) =1
2kx−PCxk
2 2
is∇h(x) =x−PCx, and therefore the gradient of the function
g(x) =1 2 kxk2 2− kx−PCx22 is∇g(x) =PCx.
10.10.3
Landweber’s Algorithm
If we select C = RJ and Q = {b}, then the CQ algorithm has the
iterative step
xk+1=xk−γAT(Axk−b), (10.53) which is the iterative step of Landweber’s algorithm. Therefore, Landwe- ber’s algorithm converges to a minimizer of the functionf(x) =kAx−bk2.
10.10.4
Projected Landweber’s Algorithm
LetC be an arbitrary closed convex set in RJ. Theprojected Landwe- ber’s algorithm has the iterative step
xk+1=PC(xk−γAT(Axk−b)). (10.54)
This also is a special case of the CQ algorithm, so we know that the pro- jected Landweber algorithm converges to a minimizer, overxin C, of the functionf(x) =kAx−bk2, whenever such minimizers exist.
10.10.5
Successive Orthogonal Projection
LetC1, ..., CI be closed convex sets inRJ, with non-empty intersection
C. Theconvex feasibility problem (CFP) is to find a member of C. Typi- cally, the orthogonal projection onto each Ci is easy to calculate, but the
orthogonal projection ontoC is not. The objective is to find a member of
C, not necessarily the orthogonal projection of x0, using the orthogonal
projections onto eachCi in turn.
For eachiletPibe the orthogonal projection operator forCiand define
T = PIPI−1· · ·P2P1. Then T is averaged. If C is non-empty, then the
members ofCare the fixed points ofT. Thesuccessive orthogonal projection
algorithm (SOP) has the iterative stepxm=Tmx0. According to the KM Theorem 10.3, the SOP sequence converges to a member ofC.
Chapter 11
Jacobi and Gauss-Seidel Methods
11.1 Chapter Summary . . . 169
11.2 The Jacobi and Gauss-Seidel Methods: An Example . . . 170
11.3 Splitting Methods . . . 170
11.4 Some Examples of Splitting Methods . . . 172
11.5 Jacobi’s Algorithm and JOR . . . 173
11.5.1 The JOR in the Nonnegative-definite Case . . . 174
11.6 The Gauss-Seidel Algorithm and SOR . . . 175
11.6.1 The Nonnegative-Definite Case . . . 175
11.6.2 The GS Algorithm as ART . . . 176
11.6.3 Successive Overrelaxation . . . 177
11.6.4 The SOR for Nonnegative-DefiniteQ . . . 178
11.7 Summary . . . 178
11.1
Chapter Summary
In this chapter we consider two well known iterative algorithms for solv- ing square systems of linear equations, the Jacobi method and the Gauss- Seidel method. Both these algorithms are easy to describe and to motivate. They both require not only that the system be square, that is, have the same number of unknowns as equations, but satisfy additional constraints needed for convergence.
Linear systemsAx=bneed not be square but can be associated with two square systems, A†Ax = A†b, the so-called normal equations, and
AA†z =b, sometimes called theBj¨orck-Elfving equations [108]. Both the Jacobi and the Gauss-Seidel algorithms can be modified to apply to any square system of linear equations, Sz =h. The resulting algorithms, the Jacobi overrelaxation (JOR) and successive overrelaxation (SOR) methods, involve the choice of a parameter. The JOR and SOR will converge for more general classes of matrices, provided that the parameter is appropriately chosen.
When we say that an iterative method is convergent, or converges, under certain conditions, we mean that it converges for any consistent system of the appropriate type, and for any starting vector; any iterative method 169
will converge if we begin at the right answer. We assume throughout this chapter thatAis anIbyJ matrix.