The function may be called by the names: c02afc, nag_zeros_poly_complex or nag_zeros_complex_poly.
c02afc attempts to find all the roots of the th degree complex polynomial equation
The roots are located using a modified form of Laguerre's method, originally proposed by Smith (1967).
The method of Laguerre (see Wilkinson (1965)) can be described by the iterative scheme
where , and is specified.
The sign in the denominator is chosen so that the modulus of the Laguerre step at , viz. , 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 function generates a sequence of iterates such that and ensures that ‘roughly’ lies inside a circular region of radius about known to contain a zero of ; that is, , where denotes the Fejér bound (see Marden (1966)) at the point . Following Smith (1967), is taken to be , where is an upper bound for the magnitude of the smallest zero given by
is the zero of smaller magnitude of the quadratic equation
and the Cauchy lower bound for the smallest zero is computed (using Newton's Method) as the positive root of the polynomial equation
Starting from the origin, successive iterates are generated according to the rule for and is ‘adjusted’ so that and . The iterative procedure terminates if is smaller in absolute value than the bound on the rounding error in and the current iterate is taken to be a zero of . The deflated polynomial of degree is then formed, and the above procedure is repeated on the deflated polynomial until , whereupon the remaining roots are obtained via the ‘standard’ closed formulae for a linear () or quadratic () equation.
Marden M (1966) Geometry of polynomials Mathematical Surveys3 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
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford
1: – IntegerInput
On entry: the degree of the polynomial, .
2: – const ComplexInput
On entry: and must contain the real and imaginary parts of (i.e., the coefficient of ), for .
3: – Nag_BooleanInput
On entry: indicates whether or not the polynomial is to be scaled. The recommended value is Nag_TRUE. See Section 9 for advice on when it may be preferable to set and for a description of the scaling strategy.
4: – ComplexOutput
On exit: the real and imaginary parts of the roots are stored in and respectively, for .
5: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, the complex variable has zero real and imaginary parts.
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
The iterative procedure has failed to converge. This error is very unlikely to occur. If it does, please contact NAG immediately, as some basic assumption for the arithmetic has been violated.
The function cannot evaluate near some of its zeros without overflow. Please contact NAG immediately.
The function cannot evaluate near some of its zeros without underflow. Please contact NAG immediately.
All roots are evaluated as accurately as possible, but because of the inherent nature of the problem complete accuracy cannot be guaranteed.
8Parallelism and Performance
c02afc is not threaded in any implementation.
If , then a scaling factor for the coefficients is chosen as a power of the base of the machine so that the largest coefficient in magnitude approaches . You should note that no scaling is performed if the largest coefficient in magnitude exceeds , even if . (, and are defined in Chapter X02.)
However, with , overflow may be encountered when the input coefficients vary widely in magnitude, particularly on those machines for which overflows. In such cases, scale should be set to Nag_FALSE and the coefficients scaled so that the largest coefficient in magnitude does not exceed .
Even so, the scaling strategy used in c02afc is sometimes insufficient to avoid overflow and/or underflow conditions. In such cases, you are recommended to scale the independent variable so that the disparity between the largest and smallest coefficient in magnitude is reduced. That is, use the function to locate the zeros of the polynomial for some suitable values of and . For example, if the original polynomial was , then choosing and , for instance, would yield the scaled polynomial , which is well-behaved relative to overflow and underflow and has zeros which are times those of .
If the function fails with NE_POLY_NOT_CONV, NE_POLY_UNFLOW or
NE_POLY_OVFLOW, 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. More
precisely, and contain the real and imaginary parts of the 1st root found,
and contain the real and imaginary parts of the 2nd root found, and so on. The real and imaginary parts of any roots not found will be set to a large negative number, specifically .
To find the roots of the polynomial , where , , , , and .