c02 Chapter Contents
c02 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zeros_real_poly (c02agc)

## 1  Purpose

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

## 2  Specification

 #include #include
 void nag_zeros_real_poly (Integer n, const double a[], Nag_Boolean scale, Complex z[], NagError *fail)

## 3  Description

nag_zeros_real_poly (c02agc) attempts to find all the roots of the $n$th degree real polynomial equation
 $Pz = a0 zn + a1 zn-1 + a2 zn-2 + ⋯ + an-1 z + an = 0 .$
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
 $Lzk = zk+1 - zk = -n P zk P′ zk ± Hzk ,$
where $H\left({z}_{k}\right)=\left(n-1\right)\left[\left(n-1\right){\left({P}^{\prime }\left({z}_{k}\right)\right)}^{2}-nP\left({z}_{k}\right){P}^{\prime \prime }\left({z}_{k}\right)\right]$ and ${z}_{0}$ is specified.
The sign in the denominator is chosen so that the modulus of the Laguerre step at ${z}_{k}$, viz. $\left|L\left({z}_{k}\right)\right|$, 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 ${z}_{1},{z}_{2},{z}_{3},\dots \text{}$, such that $\left|P\left({z}_{k+1}\right)\right|<\left|P\left({z}_{k}\right)\right|$ and ensures that ${z}_{k+1}+L\left({z}_{k+1}\right)$ ‘roughly’ lies inside a circular region of radius $\left|F\right|$ about ${z}_{k}$ known to contain a zero of $P\left(z\right)$; that is, $\left|L\left({z}_{k+1}\right)\right|\le \left|F\right|$, where $F$ denotes the Fejér bound (see Marden (1966)) at the point ${z}_{k}$. Following Smith (1967), $F$ is taken to be $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(B,1.445nR\right)$, where $B$ is an upper bound for the magnitude of the smallest zero given by
 $B = 1.0001 × min n L zk ,r1, an / a0 1/n ,$
${r}_{1}$ 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 ${z}_{k+1}={z}_{k}+L\left({z}_{k}\right)$, for $k=1,2,3,\dots \text{}$, and $L\left({z}_{k}\right)$ is ‘adjusted’ so that $\left|P\left({z}_{k+1}\right)\right|<\left|P\left({z}_{k}\right)\right|$ and $\left|L\left({z}_{k+1}\right)\right|\le \left|F\right|$. The iterative procedure terminates if $P\left({z}_{k+1}\right)$ is smaller in absolute value than the bound on the rounding error in $P\left({z}_{k+1}\right)$ and the current iterate ${z}_{p}={z}_{k+1}$ is taken to be a zero of $P\left(z\right)$ (as is its conjugate ${\stackrel{-}{z}}_{p}$ if ${z}_{p}$ is complex). The deflated polynomial $\stackrel{~}{P}\left(z\right)=P\left(z\right)/\left(z-{z}_{p}\right)$ of degree $n-1$ if ${z}_{p}$ is real ($\stackrel{~}{P}\left(z\right)=P\left(z\right)/\left(\left(z-{z}_{p}\right)\left(z-{\stackrel{-}{z}}_{p}\right)\right)$ of degree $n-2$ if ${z}_{p}$ 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.

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

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the degree of the polynomial.
Constraint: ${\mathbf{n}}\ge 1$.
2:     a[${\mathbf{n}}+1$]const doubleInput
On entry: ${\mathbf{a}}\left[\mathit{i}-1\right]$ must contain ${a}_{\mathit{i}}$ (i.e., the coefficient of ${z}^{n-\mathit{i}}$) , for $\mathit{i}=0,1,\dots ,n$.
Constraint: ${\mathbf{a}}\left[0\right]\ne 0.0$.
3:     scaleNag_BooleanInput
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 ${\mathbf{scale}}=\mathrm{Nag_FALSE}$ and for a description of the scaling strategy.
Suggested value: ${\mathbf{scale}}=\mathrm{Nag_TRUE}$.
4:     z[n]ComplexOutput
On exit: the real and imaginary parts of the roots are stored in ${\mathbf{z}}\left[\mathit{i}\right]\mathbf{.}\mathbf{re}$ and ${\mathbf{z}}\left[\mathit{i}\right]\mathbf{.}\mathbf{im}$ respectively, for $\mathit{i}=0,1,\dots ,n-1$. Complex conjugate pairs of roots are stored in consecutive pairs of z; that is, ${\mathbf{z}}\left[i+1\right]\mathbf{.}\mathbf{re}={\mathbf{z}}\left[i\right]\mathbf{.}\mathbf{re}$ and ${\mathbf{z}}\left[i+1\right]\mathbf{.}\mathbf{im}=-{\mathbf{z}}\left[i\right]\mathbf{.}\mathbf{im}$.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INTERNAL_ERROR
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.
NE_POLY_NOT_CONV
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.
NE_POLY_OVFLOW
nag_zeros_real_poly (c02agc) cannot evaluate $P\left(z\right)$ near some of its zeros without overflow. If this message occurs please contact NAG.
NE_POLY_UNFLOW
nag_zeros_real_poly (c02agc) cannot evaluate $P\left(z\right)$ near some of its zeros without underflow. If this message occurs please contact NAG.
NE_REAL_ARG_EQ
On entry, ${\mathbf{a}}\left[0\right]=0.0$.
Constraint: ${\mathbf{a}}\left[0\right]\ne 0.0$.

## 7  Accuracy

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

If ${\mathbf{scale}}=\mathrm{Nag_TRUE}$, 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 $\mathit{thresh}={b}^{{e}_{\mathrm{max}}-p}$. You should note that no scaling is performed if the largest coefficient in magnitude exceeds $\mathit{thresh}$, even if ${\mathbf{scale}}=\mathrm{Nag_TRUE}$. ($b$, ${e}_{\mathrm{max}}$ and $p$ are defined in Chapter x02.)
However, with ${\mathbf{scale}}=\mathrm{Nag_TRUE}$, 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 ${b}^{\left(4p\right)}$ 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 ${b}^{\left({e}_{\mathrm{max}}-2p\right)}$.
Even so, the scaling strategy used by 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 $\left(z\right)$ 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 $dP\left(cz\right)$ for some suitable values of $c$ and $d$. For example, if the original polynomial was $P\left(z\right)={2}^{-100}+{2}^{100}{z}^{20}$, 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 $P\left(z\right)$.
If the function fails with NE_POLY_NOT_CONV, NE_POLY_OVFLOW or NE_POLY_UNFLOW, 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 ${n}_{R}$ denote the number of roots found before the failure occurred. Then ${\mathbf{z}}\left[n-2\right]\mathbf{.}\mathbf{re}$ and ${\mathbf{z}}\left[n-2\right]\mathbf{.}\mathbf{im}$ contain the real and imaginary parts of the first root found, ${\mathbf{z}}\left[n-3\right]\mathbf{.}\mathbf{re}$ and ${\mathbf{z}}\left[n-3\right]\mathbf{.}\mathbf{im}$ contain the real and imaginary parts of the second root found, $\dots ,{\mathbf{z}}\left[n-{n}_{R}-1\right]\mathbf{.}\mathbf{re}$ and ${\mathbf{z}}\left[n-{n}_{R}-1\right]\mathbf{.}\mathbf{im}$ contain the real and imaginary parts of the ${n}_{R}$th root found. After the failure has occurred, the remaining $\left({n}_{R}\right)$ elements of z have identical real and imaginary parts, equal to the large relative number $-1.0/\left({\mathbf{nag_real_safe_small_number}}×\sqrt{2}\right)$.

## 9  Example

This example finds the roots of the fifth degree polynomial
 $z5 + 2⁢z4 + 3⁢z3 + 4⁢z2 + 5z + 6=0 .$

### 9.1  Program Text

Program Text (c02agce.c)

### 9.2  Program Data

Program Data (c02agce.d)

### 9.3  Program Results

Program Results (c02agce.r)