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

Parameters

Compulsory Input Parameters

1: $a$ – double scalar: Must contain $a$ , the coefficient of $z^{2}$ .
2: $b$ – double scalar: Must contain $b$ , the coefficient of $z$ .
3: $c$ – double scalar: Must contain $c$ , the constant coefficient.

Optional Input Parameters

None.

Output Parameters

1: $zsm (2)$ – double array: The real and imaginary parts of the smallest root in magnitude are stored in $zsm (1)$ and $zsm (2)$ respectively.
2: $zlg (2)$ – double array: The real and imaginary parts of the largest root in magnitude are stored in $zlg (1)$ and $zlg (2)$ respectively.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: On entry, $a = 0.0$ . In this case, $zsm (1)$ contains the root $- c / b$ and $zsm (2)$ contains zero.

$ifail = 2$: On entry, $a = 0.0$ and $b = 0.0$ . In this case, $zsm (1)$ contains the largest machine representable number (see nag_machine_real_largest (x02al)) and $zsm (2)$ contains zero.

$ifail = 3$: On entry, $a = 0.0$ and the root $- c / b$ overflows. In this case, $zsm (1)$ contains the largest machine representable number (see nag_machine_real_largest (x02al)) and $zsm (2)$ contains zero.

$ifail = 4$: On entry, $c = 0.0$ and the root $- b / a$ overflows. In this case, both $zsm (1)$ and $zsm (2)$ contain zero.

$ifail = 5$: On entry, $b$ is so large that $b^{2}$ is indistinguishable from $b^{2} - 4 a c$ and the root $- b / a$ overflows. In this case, $zsm (1)$ contains the root $- c / b$ and $zsm (2)$ contains zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

ifail > 0

on exit, then

zlg (1)

contains the largest machine representable number (see nag_machine_real_largest (x02al)) and

zlg (2)

contains zero.

Accuracy

ifail = 0

on exit, then the computed roots should be accurate to within a small multiple of the machine precision except when underflow (or overflow) occurs, in which case the true roots are within a small multiple of the underflow (or overflow) threshold of the machine.

Further Comments

None.

Example

This example finds the roots of the quadratic equation

z^{2} + 3 z - 10 = 0

Open in the MATLAB editor: c02aj_example

function c02aj_example


fprintf('c02aj example results\n\n');

% Roots of x^2 + 3*x - 10 = 0

a =   1;
b =   3;
c = -10;
[zsm, zlg, ifail] = c02aj(a, b, c);

disp('Roots of the quadratic equation:');

if (zsm(2) == 0) 
  % two real roots
  z(1) = zsm(1);
  z(2) = zlg(1);
else
  % two complex roots 
  z(1) = zsm(1) + i*zsm(2);
  z(2) = zlg(1) + i*zlg(2);
end
disp(z');

c02aj example results

Roots of the quadratic equation:
     2
    -5

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox