Figure 3.13 Local parameterization.
3.3.4 Singular value decomposition and the pseudo-inverse method
F or the modifications made to the continuation method described in the previous sub section an additional equation was augmented to the Newton method in order to
constrain it to search along a specified line running through the solution curve. A different approach adopted by Meijaard 1992 and Foale 1993 is to make use o f the
singular value decomposition (SVD) technique [Golub & Van Loan 1989, Press et al.
1992], which allows any matrix A to be decomposed into sub-matrices U ,W ,V , where U and V are ortho-normal matrices and W is a diagonal matrix, with positive, real elements Wj, known as the singular values o f A, such that A =U W V ^. The columns o f
U associated with the non-zero singular values form a basis for the range o f A while the columns o f V associated with the zero singular values form a basis for the nullspace
o f A.
Firstly we define a new function vector G (y)=(f(y), 0), where y = ( x ,\) and a zero has been augmented to the original equation. The Newton method applied to this system then gives
3'».! = y . - (3.26)
(a) Pseudo arclength continuation method X (b) SVD Pseudo- inverse Newton - method
Figure
3.14. A comparison of path following methods, (a) Pseudo arclength continuation; (b) SVD Pseudo-Inverse Newton method.D ,G (y J A )^ = G ( y J (3.27)
Conventionally this linear system has no well specified solution since the Jacobian
matrix is singular. In fact since the system is under-determined, it corresponds to a one
dimensional line o f solutions in the (x, X) plane (running parallel to the nullspace o f
D),G(yJ)) all o f which satisfy f(x „,\,)= 0 . However SVD allows the definition o f a
pseudo-inverse o f the matrix Dy(yJ, denoted Dy(yJ^, even in the case when it is singular, and no conventional matrix inverse exists, such that
Ay = V diag W:
(3.28)
where the 1/w, element is replaced with a zero if w, = 0 (or in a numerical implementation if it is less than a specified threshold). The solution produced by this equation has the shortest length | Ay| (i.e takes the shortest path from the origin o f the
(x, X) plane to the one-dimensional line o f solutions). This effectively restricts the
corrector iterations to a line from the predictor running perpendicular to the solution curve as shown in figure 3.14 as curve (b).
The action o f the pseudo-inverse Newton method is schematically outlined in figure 3.15. The nullspace o f D^(yJ, N u ll\D y (y J ] in the (x,X) plane, by definition, maps to the
origin o f G. Since the second component o f G(yJ is zero the range R a n g e \ D y ( y ^ ] has
basis (1, 0) in G. Given an initial guess (x„, XJ equation 3.27 defines a line o f
solutions, running parallel to V«//[D^(y„)] and passing through (x„, XJ. The SVD
solution then gives the shortest vector solution Ay from the origin to the solution line
running perpendicular to N u ll[D y(y^]. Subtracting this vector from (x„, X J in
accordance with equation 3.26, then gives the new guess (x„+;, X„+;) which also lies in
N u ll[ D y ( y J ]. This process can be repeated in an iterative manner until a satisfactory convergence to the solution point / is attained. In addition, the basis vector for the
nullspace N u l l [ D y ( f ) ] also specifies the tangent vector to the path at y*,
tj^= (dx/ds,d \ldsf. Thus since the basis vector for the nullspace is given by the vector V, SVD can also supply the tangent vector to be used in the predictor step, / = y* +
oo <1 D y G ( y o ) y = { x , X ) ( Xq, Xq) R a n g e [ D ^ G ( v o ) ] SVD solution of D .G ( y„ ) Ay = G ( Vo ) Line of solutions of D y G ( y o ) Ay = G ( y Q )
Figure 3.15. A schematic outline of the SVD pseudo-inverse Newton method. The Newton method with initial guess (xo,Xo), applied to the augmented system G(y), defines a line of solutions DyG(yo)Ay = G(yo) of which SVD finds the closest to the origin. This vector when applied to (xo,Xo) produces an improved approximation of the root satisfying f(x, X)=0.
As where Aj is now interpreted as the step length along the path. Constrained in this way any fold point on the path can be negotiated without problem.
The above approach provides an elegant formulation for the path following method and is the one adopted for the numerical studies performed here. The step length As control used, is a simple adaptive scheme based on the specification o f an ideal number o f
Newton iterations required for convergence. If the actual number o f iterations required is greater than this number then the step-size is halved; if it is less than the specified
number then the step size is multiplied by a factor o f 1.5. This control mechanism has
proven to be adequate for the range o f applications tackled here.
Before leaving this section we note that the path following methods described here for obtaining the bifurcation diagram % vs X specified by f(r,X )= 0 , can easily be extended
(in a similar manner to the Newton method discussed in sub-section 3.3.1) for
application to systems o f equations, thus allowing the varieties defined in section 3.2 to be traced out numerically. Similarly augmenting additional equations characteristic o f bifurcation (e.g. fold) points allows the paths o f the bifurcation lines to be traced
numerically. Path following methods can also be generalised for numerical investigation o f bifurcational behaviour in partial differential equations (see for example Kubikec and M arek [1983] or W inters [1991]).