polyroots_module Module

length of the vector a

vector of double

integers such that il<i<ir

integers such that il<i<ir

integers such that il<i<ir

vector of coefficients in order of decreasing powers

degree of polynomial

output vector of real parts of the zeros

output vector of imaginary parts of the zeros

status output:

coefficients

real components of roots

imaginary components of roots

coefficients

real components of roots

imaginary components of roots

coefficients

coefficients

coefficients

real part of first root

imaginary part of first root

real part of second root

imaginary part of second root

degree of the monic polynomial.

coefficients of the polynomial. in order of decreasing powers.

real part of output roots

imaginary part of output roots

return code:

accuracy hint.

Degree of the polynomial

Number of derivatives to be calculated:

vector containing the N+1 coefficients of polynomial. A(I) = coefficient of Z**(N+1-I)

point at which the evaluation is to take place

Array of 2(M+1) words: C(I+1) contains the complex value of the I-th derivative at Z, I=0,...,M

Array of 2(M+1) words: B(I) contains the bounds on the real and imaginary parts of C(I) if they were requested. only needed if bounds are to be calculated. It is not used otherwise.

A logical variable, e.g. .TRUE. or .FALSE. which is to be set .TRUE. if bounds are to be computed.

N, the degree of P(Z)

complex vector containing coefficients of P(Z), A(I) = coefficient of Z**(N+1-i)

N word complex vector. On input: containing initial estimates for zeros if these are known. On output: Ith zero

flag to indicate if initial estimates of zeros are input:

an N word array. S(I) = bound for R(I)

degree of P(X)

real vector containing coefficients of P(X), A(I) = coefficient of X**(N+1-I)

N word complex vector. On Input: containing initial estimates for zeros if these are known. On output: ith zero.

flag to indicate if initial estimates of zeros are input:

an N word array. bound for R(I).

degree of polynomial

NDEG+1 coefficients in descending order. i.e., P(Z) = COEFF(1)*(Z**NDEG) + COEFF(NDEG)*Z + COEFF(NDEG+1)

NDEG vector of roots

Output Error Code

must be set to the row dimension of two-dimensional array parameters as declared in the calling program dimension statement.

order of the matrix

low and igh are integers determined by the balancing subroutine balanc. if balanc has not been used, set low=1, igh=n.

low and igh are integers determined by the balancing subroutine balanc. if balanc has not been used, set low=1, igh=n.

On input: contains the upper hessenberg matrix. information about the transformations used in the reduction to hessenberg form by elmhes or orthes, if performed, is stored in the remaining triangle under the hessenberg matrix.

the real parts of the eigenvalues. the eigenvalues are unordered except that complex conjugate pairs of values appear consecutively with the eigenvalue having the positive imaginary part first. if an error exit is made, the eigenvalues should be correct for indices ierr+1,...,n.

the imaginary parts of the eigenvalues. the eigenvalues are unordered except that complex conjugate pairs of values appear consecutively with the eigenvalue having the positive imaginary part first. if an error exit is made, the eigenvalues should be correct for indices ierr+1,...,n.

is set to:

polynomial degree

polynomial coefficients array, in order of decreasing powers

real parts of roots

imaginary parts of roots

output from the lapack solver.

polynomial degree

polynomial coefficients array, in order of decreasing powers

roots

output from the lapack solver.

degree of polynomial

(NDEG+1) coefficients in descending order. i.e., P(Z)= COEFF(1)*(Z**NDEG) + COEFF(NDEG)*Z + COEFF(NDEG+1)

(NDEG) vector of roots

Output Error Code.

row dimension of two-dimensional array parameters

the order of the matrix

integer determined by the balancing subroutine cbal. if cbal has not been used, set low=1

integer determined by the balancing subroutine cbal. if cbal has not been used, igh=n.

On input: hr and hi contain the real and imaginary parts, respectively, of the complex upper hessenberg matrix. their lower triangles below the subdiagonal contain information about the unitary transformations used in the reduction by corth, if performed.

See hr description

the real parts of the eigenvalues. if an error exit is made, the eigenvalues should be correct for indices ierr+1,...,n.

the imaginary parts of the eigenvalues. if an error exit is made, the eigenvalues should be correct for indices ierr+1,...,n.

is set to:

degree of polynomial

vector of coefficients in order of decreasing powers

real part of the zero

imaginary part of the zero

convergence tolerance for the root

convergence tolerance for x

maximum number of iterations

status flag:

degree of polynomial

vector of coefficients in order of decreasing powers

real part of the zero

imaginary part of the zero

convergence tolerance for the root

convergence tolerance for x

maximum number of iterations

status flag:

degree of the polynomial and size of 'roots' array

coeffs of the polynomial, in order of increasing powers.

array which will hold all roots that had been found. If the flag 'use_roots_as_starting_points' is set to .true., then instead of point (0,0) we use value from this array as starting point for cmplx_laguerre

after all roots have been found by dividing original polynomial by each root found, you can opt in to polish all roots using full polynomial. [default is false]

usually we start Laguerre's method from point (0,0), but you can decide to use the values of 'roots' array as starting point for each new root that is searched for. This is useful if you have very rough idea where some of the roots can be. [default is false]

vectors of real parts of the coefficients in order of decreasing powers.

vectors of imaginary parts of the coefficients in order of decreasing powers.

degree of polynomial

real parts of the zeros

imaginary parts of the zeros

true only if leading coefficient is zero or if cpoly has found fewer than degree zeros.

degree of the polynomial.

complex vector of n+1 components, poly(i) is the coefficient of x**(i-1), i=1,...,n+1 of the polynomial p(x)

the max number of allowed iterations.

complex vector of n components, containing the approximations to the roots of p(x).

real vector of n components, containing the error bounds to the approximations of the roots, i.e. the disk of center root(i) and radius radius(i) contains a root of p(x), for i=1,...,n. radius(i) is set to -1 if the corresponding root cannot be represented as floating point due to overflow or underflow.

vector of n components detecting an error condition:

degree of the polynomial p(x)

coefficients of the polynomial p(x)

upper bounds on the backward perturbations on the coefficients of p(x) when applying ruffini-horner's rule

upper bounds on the backward perturbations on the coefficients of p(x) when applying (2), (2')

value at which the newton correction is computed

the min positive real(wp), small=2**(-1074) for the ieee.

upper bound to the distance of z from the closest root of the polynomial computed according to (4).

newton's correction

this variable is .true. if the computed value p(z) is reliable, i.e., (3) is not satisfied in z. again is .false., otherwise.

degree of the polynomial

index of the component of root with respect to which the aberth correction is computed

vector containing the current approximations to the roots

aberth's correction (compare (1))

number of the coefficients of the polynomial

moduli of the coefficients of the polynomial

starting approximations

if a component is -1 then the corresponding root has a too big or too small modulus in order to be represented as double float with no overflow/underflow

number of roots which cannot be represented without overflow/underflow

the min positive real(wp), small=2**(-1074) for the ieee.

the max real(wp), big=2**1023 for the ieee standard.

length of the vector h

vector of logical

integer

maximum integer such that il<i, h(il)=.true.

length of the vector h

vector of logical

integer

minimum integer such that ir>i, h(ir)=.true.

length of the vector a

vector defining the points (j,a(j))

abscissa of the common vertex of the two sets

the number of elements of each set is m+1

vector defining the vertices of the convex hull, i.e., h(j) is .true. if (j,a(j)) is a vertex of the convex hull this vector is used also as output.

coefficients

polynomial degree

the root approximations

the backward error in each approximation

the condition number of each root approximation

vector of coefficients in order of decreasing powers

output vector of real parts of the zeros

output vector of imaginary parts of the zeros

the real parts to be sorted. on exit,x has been sorted into increasing order (x(1) <= ... <= x(n))

the imaginary parts to be sorted

Degree of the polynomial

Contains the coefficients of a polynomial, high order coefficient first with A(1)/=0.

Contains the real parts of the roots

Contains the imaginary parts of the roots

Error flag:

contains the complex coefficients of a polynomial high order coefficient first, with a([,]1)/=0. the real and imaginary parts of the jth coefficient must be provided in a([],j). the contents of this array will not be modified by the subroutine.

degree of the polynomial

contains the polynomial roots stored as complex numbers. the real and imaginary parts of the jth root will be stored in z([*,]j).

error flag. set by the subroutine to 0 on normal termination. set to -1 if a([*,]1)=0. set to -2 if ndeg <= 0. set to j > 0 if the iteration count limit has been exceeded and roots 1 through j have not been determined.

the row dimension of two-dimensional array parameters as declared in the calling program dimension statement

the order of the matrix

low and igh are integers determined by the balancing subroutine cbal. if cbal has not been used, set low=1, igh=n

low and igh are integers determined by the balancing subroutine cbal. if cbal has not been used, set low=1, igh=n

see hi description

Input: hr and hi contain the real and imaginary parts, respectively, of the complex upper hessenberg matrix. their lower triangles below the subdiagonal contain information about the unitary transformations used in the reduction by corth, if performed.

the real and imaginary parts, respectively, of the eigenvalues. if an error exit is made, the eigenvalues should be correct for indices ierr+1,...,n,

is set to: