NAG Library Routine Document
C02AFF finds all the roots of a complex polynomial equation, using a variant of Laguerre's method.
||A(2,N+1), Z(2,N), W(4*(N+1))
C02AFF 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
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 routine generates a sequence of iterates
, such that
and ensures that
‘roughly’ lies inside a circular region of radius
known to contain a zero of
; that is,
denotes the Fejér bound (see Marden (1966)
) at the point
. Following Smith (1967)
is taken to be
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
is ‘adjusted’ so that
. 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.
Note that C02AHF
can be used to obtain the roots of a quadratic, cubic (
) and quartic (
) polynomial, respectively.
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
- 1: A(,) – REAL (KIND=nag_wp) arrayInput
: if A
is declared with bounds
must contain the real and imaginary parts of
(i.e., the coefficient of
- 2: N – INTEGERInput
On entry: , the degree of the polynomial.
- 3: SCAL – LOGICALInput
: indicates whether or not the polynomial is to be scaled. See Section 8
for advice on when it may be preferable to set
and for a description of the scaling strategy.
- 4: Z(,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the real and imaginary parts of the roots are stored in
and respectively, for .
- 5: W() – REAL (KIND=nag_wp) arrayWorkspace
- 6: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|On entry,|| and ,|
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
Either overflow or underflow prevents the evaluation of
near some of its zeros. This error is very unlikely to occur. If it does, please contact NAG
. See also Section 8
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
, 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
are defined in Chapter X02
, overflow may be encountered when the input coefficients
vary widely in magnitude, particularly on those machines for which
overflows. In such cases, SCAL
should be set to .FALSE. and the coefficients scaled so that the largest coefficient in magnitude does not exceed
Even so, the scaling strategy used by C02AFF 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 routine 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 routine fails with
, 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.
denote the number of roots found before the failure occurred. Then
contain the real and imaginary parts of the first root found,
contain the real and imaginary parts of the second root found,
contain the real and imaginary parts of the
th root found. After the failure has occurred, the remaining
elements of Z
contain a large negative number (equal to
For this routine two examples are presented. There is a single example program for C02AFF, with a main program and the code to solve the two example problems given in the subroutines EX1 and EX2.
Example 1 (EX1)
This example finds the roots of the polynomial
Example 2 (EX2)
This example solves the same problem as subroutine EX1, but in addition attempts to estimate the accuracy of the computed roots using a perturbation analysis. Further details can be found in Thompson (1991)
9.1 Program Text
Program Text (c02affe.f90)
9.2 Program Data
Program Data (c02affe.d)
9.3 Program Results
Program Results (c02affe.r)