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15 Orthogonal Polynomials: General Aspects
The polynomial normalization factor is
√
2n + 2. The radial polynomials can be
evaluated with the SPECFUNPHYS program zernike with syntax [Z, pn,
pint, info] = zernike(m, nmax). The input variable “m” is an integer
as defined above and “nmax” the maximum n-value under consideration. The output
variable “Z” is the radial Zernike polynomial, “pn” the normalization factor, “pint”
the integration interval for normalization, and “info” some general information. The
following example shows how to generate an object of the superclass polymeth
and below how to use the methods of polymeth.
m=1; nmax = 7; % input variables for function zernike
[Z,pn,pint,info] = zernike(m, nmax); % polynom
obj = orthpoly(pc,pn,pint,info);
15.3.2 The Class polymeth
polymeth comes with the following methods available for orthogonal polynomial
objects:
Method plot with syntax plot(obj, n, x, ha). “obj” is the object, all
other inputs are optional. “n” (n ≥ 0) is an integer vector whose elements determine
the order of the polynomials to be plotted. The default is that all polynomials are
plotted. “x” is the region or the polynomial arguments at which the polynomial shall
be plotted. For example, for x = [x1, x2] polynomial will be evaluated at 100 evenly
spaced points between x1 and x2; in case “wint” has n components the polynomials
will be computed at these n values. The default is the property “polyint”. The last
input variable “ah” is an axes handle object that will be used for plotting.
Method deri with syntax Pder = deri(obj, n) computes the n-th derivative of the orthogonal polynomials.
Method polyvalue with syntax [res, n] = polyvalue(obj,x,n). “x”
and “n” are optional inputs. “x” is the position at which the polynomials shall be
evaluated with the same rules as for method plot. “n” is an integer vector larger
than or equal to 1 of the polynomial rows (obj.polycoef) which shall be evaluated.
Method opolyprod for computing the product of orthogonal polynomials
with syntax [Cprod, prodtab, info] = opolyprod(obj1, ...).
For ... = opolyprod(obj) all possible polynomial products will be
computed. Additional arguments could be a 2nd object, which must be the second
input or name–value pairs. ... = opolyprod(obj1, obj2) all possible
combination between object “obj1” and “obj2” will be computed. In case there
is no second object as input, “obj2” will be equal to “obj1”. opolyprod(_,
’n1’, [0,1,4]) all possible combinations of the polynomials 0, 1, 4 of obj1
with “obj2” will be computed. opolyprod(_, ’n2’, [0,1,4]) the same
but vice versa. The outputs are “Cprod”, the product polynomials, “prodtab” the
15 Orthogonal Polynomials: General Aspects
The polynomial normalization factor is
√
2n + 2. The radial polynomials can be
evaluated with the SPECFUNPHYS program zernike with syntax [Z, pn,
pint, info] = zernike(m, nmax). The input variable “m” is an integer
as defined above and “nmax” the maximum n-value under consideration. The output
variable “Z” is the radial Zernike polynomial, “pn” the normalization factor, “pint”
the integration interval for normalization, and “info” some general information. The
following example shows how to generate an object of the superclass polymeth
and below how to use the methods of polymeth.
m=1; nmax = 7; % input variables for function zernike
[Z,pn,pint,info] = zernike(m, nmax); % polynom
obj = orthpoly(pc,pn,pint,info);
15.3.2 The Class polymeth
polymeth comes with the following methods available for orthogonal polynomial
objects:
Method plot with syntax plot(obj, n, x, ha). “obj” is the object, all
other inputs are optional. “n” (n ≥ 0) is an integer vector whose elements determine
the order of the polynomials to be plotted. The default is that all polynomials are
plotted. “x” is the region or the polynomial arguments at which the polynomial shall
be plotted. For example, for x = [x1, x2] polynomial will be evaluated at 100 evenly
spaced points between x1 and x2; in case “wint” has n components the polynomials
will be computed at these n values. The default is the property “polyint”. The last
input variable “ah” is an axes handle object that will be used for plotting.
Method deri with syntax Pder = deri(obj, n) computes the n-th derivative of the orthogonal polynomials.
Method polyvalue with syntax [res, n] = polyvalue(obj,x,n). “x”
and “n” are optional inputs. “x” is the position at which the polynomials shall be
evaluated with the same rules as for method plot. “n” is an integer vector larger
than or equal to 1 of the polynomial rows (obj.polycoef) which shall be evaluated.
Method opolyprod for computing the product of orthogonal polynomials
with syntax [Cprod, prodtab, info] = opolyprod(obj1, ...).
For ... = opolyprod(obj) all possible polynomial products will be
computed. Additional arguments could be a 2nd object, which must be the second
input or name–value pairs. ... = opolyprod(obj1, obj2) all possible
combination between object “obj1” and “obj2” will be computed. In case there
is no second object as input, “obj2” will be equal to “obj1”. opolyprod(_,
’n1’, [0,1,4]) all possible combinations of the polynomials 0, 1, 4 of obj1
with “obj2” will be computed. opolyprod(_, ’n2’, [0,1,4]) the same
but vice versa. The outputs are “Cprod”, the product polynomials, “prodtab” the