NAG Library Routine Document

C02AGF

The sign in the denominator is chosen so that the modulus of the Laguerre step at zk, viz. Lzk, is as small as possible. The method can be shown to be cubically convergent for isolated roots (real or complex) and linearly convergent for multiple roots.

The routine generates a sequence of iterates z1,z2,z3,…, such that Pzk+1<Pzk and ensures that zk+1+Lzk+1 ‘roughly’ lies inside a circular region of radius F about zk known to contain a zero of Pz; that is, Lzk+1≤F, where F denotes the Fejér bound (see Marden (1966)) at the point zk. Following Smith (1967), F is taken to be minB,1.445nR , where B is an upper bound for the magnitude of the smallest zero given by

B = 1.0001 × min n L zk ,r1, an / a0 1/n ,

r1 is the zero X of smaller magnitude of the quadratic equation

P′′ zk 2 n n-1 X2 + P′ zk n X + 12 Pzk = 0

and the Cauchy lower bound R for the smallest zero is computed (using Newton's Method) as the positive root of the polynomial equation

a0 zn + a1 zn-1 + a2 zn-2 +⋯+ an-1 z - an = 0 .

Starting from the origin, successive iterates are generated according to the rule zk+1 = zk + L zk , for k=1 , 2 , 3 ,…, and Lzk is ‘adjusted’ so that P zk+1 < P zk and L zk+1 ≤ F . The iterative procedure terminates if P zk+1 is smaller in absolute value than the bound on the rounding error in P zk+1 and the current iterate zp = zk+1 is taken to be a zero of Pz (as is its conjugate z-p if zp is complex). The deflated polynomial P~ z = P z / z-zp of degree n-1 if zp is real ( P~ z = P z / z-zp z-z-p of degree n-2 if zp is complex) is then formed, and the above procedure is repeated on the deflated polynomial until n<3, whereupon the remaining roots are obtained via the ‘standard’ closed formulae for a linear (n=1) or quadratic (n=2) equation.

Note that C02AJF, C02AKF and C02ALF can be used to obtain the roots of a quadratic, cubic (n=3) and quartic (n=4) polynomial, respectively.

4 References

Marden M (1966) Geometry of polynomials Mathematical Surveys 3 American Mathematical Society, Providence, RI

Smith B T (1967) ZERPOL: A zero finding algorithm for polynomials using Laguerre's method Technical Report Department of Computer Science, University of Toronto, Canada

Thompson K W (1991) Error analysis for polynomial solvers Fortran Journal (Volume 3) 3 10–13

Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

5 Parameters

1: A(N+1) – double precision arrayInput: On entry: if A is declared with bounds 0:N, then Ai must contain ai (i.e., the coefficient of zn-i ), for i=0,1,…,n .
Constraint: A0 ≠ 0.0 .
2: N – INTEGERInput: On entry: n, the degree of the polynomial.
Constraint: N≥1.
3: SCAL – LOGICALInput: On entry: indicates whether or not the polynomial is to be scaled. See Section 8 for advice on when it may be preferable to set SCAL=.FALSE. and for a description of the scaling strategy.

Suggested value: SCAL=.TRUE..
4: Z(2,N) – double precision arrayOutput: On exit: the real and imaginary parts of the roots are stored in Z1i and Z2i respectively, for i=1,2,…,n . Complex conjugate pairs of roots are stored in consecutive pairs of elements of Z; that is, Z1i+1 = Z1i and Z2i+1 = -Z2i .
5: W(2×N+1) – double precision arrayWorkspace
6: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to 0, -1 or 1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

On exit: IFAIL=0 unless the routine detects an error (see Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1 or 1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.

6 Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL=1: On entry, A0=0.0,
or N<1.

IFAIL=2: The iterative procedure has failed to converge. This error is very unlikely to occur. If it does, please contact NAG, as some basic assumption for the arithmetic has been violated. See also Section 8.

IFAIL=3: Either overflow or underflow prevents the evaluation of Pz near some of its zeros. This error is very unlikely to occur. If it does, please contact NAG. See also Section 8.

7 Accuracy

All roots are evaluated as accurately as possible, but because of the inherent nature of the problem complete accuracy cannot be guaranteed. See also Section 9.

8 Further Comments

If SCAL=.TRUE., then a scaling factor for the coefficients is chosen as a power of the base b of the machine so that the largest coefficient in magnitude approaches thresh=bemax-p. You should note that no scaling is performed if the largest coefficient in magnitude exceeds thresh, even if SCAL=.TRUE.. (b, emax and p are defined in Chapter X02.)

However, with SCAL=.TRUE., overflow may be encountered when the input coefficients a0,a1,a2,…,an vary widely in magnitude, particularly on those machines for which b4p overflows. In such cases, SCAL should be set to .FALSE. and the coefficients scaled so that the largest coefficient in magnitude does not exceed b emax-2p .

Even so, the scaling strategy used by C02AGF is sometimes insufficient to avoid overflow and/or underflow conditions. In such cases, you are recommended to scale the independent variable z so that the disparity between the largest and smallest coefficient in magnitude is reduced. That is, use the routine to locate the zeros of the polynomial dPcz for some suitable values of c and d. For example, if the original polynomial was Pz=2-100+2100z20, then choosing c=2-10 and d=2100, for instance, would yield the scaled polynomial 1+z20, which is well-behaved relative to overflow and underflow and has zeros which are 210 times those of Pz.

If the routine fails with IFAIL=2 or 3, then the real and imaginary parts of any roots obtained before the failure occurred are stored in Z in the reverse order in which they were found. Let nR denote the number of roots found before the failure occurred. Then Z1n and Z2n contain the real and imaginary parts of the first root found, Z1n-1 and Z2n-1 contain the real and imaginary parts of the second root found, … , Z1 n- nR+1 and Z2 n- nR+1 contain the real and imaginary parts of the nR th root found. After the failure has occurred, the remaining 2× n-nR elements of Z contain a large negative number (equal to -1 / X02AMF×2 ).