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Find zeros of function in given interval


z = fnzeros(f,[a b])
z = fnzeros(f)


z = fnzeros(f,[a b]) is an ordered list of the zeros of the univariate spline f in the interval [a .. b] .

z = fnzeros(f) is a list of the zeros in the basic interval of the spline f.

A spline zero is either a maximal closed interval over which the spline is zero, or a zero crossing (a point across which the spline changes sign).

The list of zeros, z, is a matrix with two rows. The first row is the left endpoint of the intervals and the second row is the right endpoint. Each column z(:,j) contains the left and right endpoint of a single interval.

These intervals are of three kinds:

  • If the endpoints are different, then the function is zero on the entire interval. In this case the maximal interval is given, regardless of knots that may be in the interior of the interval.

  • If the endpoints are the same and coincident with a knot, then the function in f has a zero at that point. The spline could cross zero, touch zero or be discontinuous at this point.

  • If the endpoints are the same and not coincident with a knot, then the spline has a zero crossing at this point.

If the spline, f, touches zero at a point that is not a knot, but does not cross zero, then this zero may not be found. If it is found, then it may be found twice.


Example 1. The following code constructs and plots a piecewise linear spline that has each of the three kinds of zeros: touch zero, cross zero, and zero for an interval. fnzeroscomputes all the zeros, and then the code plots the results on the graph.

sp = spmak(augknt(1:7,2),[1,0,1,-1,0,0,1]);
z = fnzeros(sp)
nz = size(z,2);
hold on
plot(z(1,:),zeros(1,nz),'>',z(2,:),zeros(1,nz),'<'), hold off

This gives the following list of zeros:

z =
    2.0000    3.5000    5.0000
    2.0000    3.5000    6.0000

In this simple example, even for the second kind of zero, the two endpoints agree to all places.

Example 2. The following code generates and plots a spline function with many extrema and locates all extrema by computing the zeros of the spline function's first derivative there.

f = spmak( 1:21, rand( 1, 15 )-0.5 ); 
interval = fnbrk( f, 'interval' );
z = fnzeros( fnder( f ) );
z = z(1,:);
values = fnval( f, [interval, z] );
min( values )
hold on
hold off

Your plot will be different to the example following because of the use of rand to generate random coefficients.

Example 3. We construct a spline with a zero at a jump discontinuity and in B-form and find all the spline's zeros in an interval that goes beyond its basic interval.

sp = spmak([0 0 1 1 2],[1 0 -.2]);
z = fnzeros(sp,[.5, 2.7])
zy = zeros(1,size(z,2));
hold on, plot(z(1,:),zy,'>',z(2,:),zy,'<'), hold off

This gives the following list of zeros:

z =
    1.0000    2.0000
    1.0000    2.7000

Notice the resulting zero interval [2..2.7], due to the fact that, by definition, a spline in B-form is identically zero outside its basic interval, [0..2].

Example 4. The following example shows the use of fnzeros with a discontinuous function. The following code creates and plots a discontinuous piecewise linear function, and finds the zeros.

 sp = spmak([0 0 1 1 2 2],[-1 1 -1 1]);

This gives the following list of zeros, in (1..2) and (0..1) and the jump through zero at 1:

ans =

    0.5000    1.0000    1.5000
    0.5000    1.0000    1.5000


fnzeros first converts the function to B-form. The function performs some preprocessing to handle discontinuities, and then uses the algorithm of Mørken and Reimers.

Reference: Knut Mørken and Martin Reimers, An unconditionally convergent method for computing zeros of splines and polynomials, Math. Comp. 76:845--865, 2007.

See Also