nag_zeros_real_poly (c02agc) : NAG Library, Mark 24

nag_zeros_real_poly (c02agc) finds all the roots of a real polynomial equation, using a variant of Laguerre's method.

nag_zeros_real_poly (c02agc) attempts to find all the roots of the

n

th degree real polynomial equation

P (z) = a_{0} z^{n} + a_{1} z^{n - 1} + a_{2} z^{n - 2} + \dots + a_{n - 1} z + a_{n} = 0 .

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

L (z_{k}) = z_{k + 1} - z_{k} = \frac{- n P (z_{k})}{P^{'} (z_{k}) \pm \sqrt{H (z_{k})}},

where

H (z_{k}) = (n - 1) [(n - 1) {(P^{'} (z_{k}))}^{2} - n P (z_{k}) P^{''} (z_{k})]

, and

z_{0}

is specified.

The sign in the denominator is chosen so that the modulus of the Laguerre step at

z_{k}

, viz.

|L (z_{k})|

, 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

z_{1}, z_{2}, z_{3}, \dots,

such that

|P (z_{k + 1})| < |P (z_{k})|

and ensures that

z_{k + 1} + L (z_{k + 1})

‘roughly’ lies inside a circular region of radius

|F|

about

z_{k}

known to contain a zero of

P (z)

; that is,

|L (z_{k + 1})| \leq |F|

, where

F

denotes the Fejér bound (see Marden (1966)) at the point

z_{k}

. Following Smith (1967),

F

is taken to be

\min (B, 1.445 n R)

, where

B

is an upper bound for the magnitude of the smallest zero given by

B = 1.0001 \times \min (\sqrt{n} L (z_{k}), |r_{1}|, {|a_{n} / a_{0}|}^{1 / n}),

r_{1}

is the zero

X

of smaller magnitude of the quadratic equation

(P^{''} (z_{k}) / (2 n (n - 1))) X^{2} + (P^{'} (z_{k}) / n) X + \frac{1}{2} P (z_{k}) = 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

|a_{0}| z^{n} + |a_{1}| z^{n - 1} + |a_{2}| z^{n - 2} + \dots + |a_{n - 1}| z - |a_{n}| = 0 .

Starting from the origin, successive iterates are generated according to the rule

z_{k + 1} = z_{k} + L (z_{k})

, for

k = 1, 2, 3, \dots

, and

L (z_{k})

is ‘adjusted’ so that

|P (z_{k + 1})| < |P (z_{k})|

and

|L (z_{k + 1})| \leq |F|

. The iterative procedure terminates if

P (z_{k + 1})

is smaller in absolute value than the bound on the rounding error in

P (z_{k + 1})

and the current iterate

z_{p} = z_{k + 1}

is taken to be a zero of

P (z)

(as is its conjugate

{\bar{z}}_{p}

z_{p}

is complex). The deflated polynomial

\tilde{P} (z) = P (z) / ({z - z}_{p})

of degree

n - 1

z_{p}

is real (

\tilde{P} (z) = P (z) / (({z - z}_{p}) (z - {\bar{z}}_{p}))

of degree

n - 2

z_{p}

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.

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

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

On entry:

n

, the degree of the polynomial.

Constraint:

n \geq 1

On entry:

a [i]

must contain

a_{i}

(i.e., the coefficient of

z^{n - i}

), for

i = 0, 1, \dots, n

Constraint:

a [0] \neq 0.0

On entry: indicates whether or not the polynomial is to be scaled. See Section 9 for advice on when it may be preferable to set

scale = Nag_FALSE

and for a description of the scaling strategy.

Suggested value:

scale = Nag_TRUE

On exit: the real and imaginary parts of the roots are stored in

z [i] . re

and

z [i] . im

respectively, for

i = 0, 1, \dots, n - 1

. Complex conjugate pairs of roots are stored in consecutive pairs of z; that is,

z [i + 1] . re = z [i] . re

and

z [i + 1] . im = - z [i] . im

The NAG error argument (see Section 3.6 in the Essential Introduction).

Dynamic memory allocation failed.

On entry, argument

⟨value⟩

had an illegal value.

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 1

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

P (z)

near some of its zeros without overflow. Please contact NAG immediately.

The function cannot evaluate

P (z)

near some of its zeros without underflow. Please contact NAG immediately.

On entry,

a [0] = ⟨value⟩

.
Constraint:

a [0] \neq 0.0

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

scale = Nag_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 = b^{e_{\max} - p}

. You should note that no scaling is performed if the largest coefficient in magnitude exceeds

thresh

, even if

scale = Nag_TRUE

. (

b

e_{\max}

and

p

are defined in Chapter x02.)

However, with

scale = Nag_TRUE

, overflow may be encountered when the input coefficients

a_{0}, a_{1}, a_{2}, \dots, a_{n}

vary widely in magnitude, particularly on those machines for which

b^{4 p}

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

b^{e_{\max} - 2 p}

Even so, the scaling strategy used in nag_zeros_real_poly (c02agc) 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 function to locate the zeros of the polynomial

d \times P (c z)

for some suitable values of

c

and

d

. For example, if the original polynomial was

P (z) = 2^{- 100} + 2^{100} z^{20}

, then choosing

c = 2^{- 10}

and

d = 2^{100}

, for instance, would yield the scaled polynomial

1 + z^{20}

, which is well-behaved relative to overflow and underflow and has zeros which are

2^{10}

times those of

P (z)

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,

z [n - 1] . re

and

z [n - 1] . im

contain the real and imaginary parts of the 1st root found,

z [n - 2] . re

and

z [n - 2] . im

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

- 1.0 / (\sqrt{2.0} \times nag_real_safe_small_number)

To find the roots of the 5th degree polynomial

z^{5} + 2 z^{4} + 3 z^{3} + 4 z^{2} + 5 z + 6 = 0

NAG Library Function Document

nag_zeros_real_poly (c02agc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Documentnag_zeros_real_poly (c02agc)