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23.1 Introduction to fast Fourier transform | ||
23.2 Functions and Variables for fast Fourier transform | ||
23.3 Introduction to Fourier series | ||
23.4 Functions and Variables for Fourier series |
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The fft
package comprises functions for the numerical (not symbolic)
computation of the fast Fourier transform.
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Translates complex values of the form r %e^(%i t)
to the form
a + b %i
, where r is the magnitude and t is the phase.
r and t are 1-dimensional arrays of the same size.
The array size need not be a power of 2.
The original values of the input arrays are
replaced by the real and imaginary parts, a
and b
, on return.
The outputs are calculated as
a = r cos(t) b = r sin(t)
polartorect
is the inverse function of recttopolar
.
load(fft)
loads this function. See also fft
.
Translates complex values of the form a + b %i
to the form
r %e^(%i t)
, where a is the real part and b is the imaginary
part. a and b are 1-dimensional arrays of the same size.
The array size need not be a power of 2.
The original values of the input arrays are
replaced by the magnitude and angle, r
and t
, on return.
The outputs are calculated as
r = sqrt(a^2 + b^2) t = atan2(b, a)
The computed angle is in the range -%pi
to %pi
.
recttopolar
is the inverse function of polartorect
.
load(fft)
loads this function. See also fft
.
Computes the inverse complex fast Fourier transform. y is a list or
array (named or unnamed) which contains the data to transform. The number of
elements must be a power of 2. The elements must be literal numbers (integers,
rationals, floats, or bigfloats) or symbolic constants, or expressions
a + b*%i
where a
and b
are literal numbers or symbolic
constants.
inverse_fft
returns a new object of the same type as y,
which is not modified.
Results are always computed as floats
or expressions a + b*%i
where a
and b
are floats.
The inverse discrete Fourier transform is defined as follows.
Let x
be the output of the inverse transform.
Then for j
from 0 through n - 1
,
x[j] = sum(y[k] exp(+2 %i %pi j k / n), k, 0, n - 1)
load(fft)
loads this function.
See also fft
(forward transform), recttopolar
, and
polartorect
.
Examples:
Real data.
(%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $ (%i4) L1 : inverse_fft (L); (%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0, 4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284] (%i5) L2 : fft (L1); (%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i, 4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0, 7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0] (%i6) lmax (abs (L2 - L)); (%o6) 3.545L-16
Complex data.
(%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $ (%i4) L1 : inverse_fft (L); (%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0, - 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0, 2.828 %i + 2.828] (%i5) L2 : fft (L1); (%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, 1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 1.0 %i + 1.0, 1.0 - 7.368L-20 %i] (%i6) lmax (abs (L2 - L)); (%o6) 6.841L-17
Computes the complex fast Fourier transform. x is a list or array
(named or unnamed) which contains the data to transform. The number of
elements must be a power of 2. The elements must be literal numbers (integers,
rationals, floats, or bigfloats) or symbolic constants, or expressions
a + b*%i
where a
and b
are literal numbers or symbolic
constants.
fft
returns a new object of the same type as x,
which is not modified.
Results are always computed as floats
or expressions a + b*%i
where a
and b
are floats.
The discrete Fourier transform is defined as follows.
Let y
be the output of the transform.
Then for k
from 0 through n - 1
,
y[k] = (1/n) sum(x[j] exp(-2 %i %pi j k / n), j, 0, n - 1)
When the data x are real,
real coefficients a
and b
can be computed such that
x[j] = sum (a[k] * cos (2*%pi*j*k / n) + b[k] * sin (2*%pi*j*k / n), k, 0, n/2)
with
a[0] = realpart (y[0]) b[0] = 0
and, for k from 1 through n/2 - 1,
a[k] = realpart (y[k] + y[n - k]) b[k] = imagpart (y[n - k] - y[k])
and
a[n/2] = realpart (y[n/2]) b[n/2] = 0
load(fft)
loads this function.
See also inverse_fft
(inverse transform), recttopolar
, and
polartorect
.
Examples:
Real data.
(%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $ (%i4) L1 : fft (L); (%o4) [0.0, - 1.811 %i - .1036, 0.0, .6036 - .3107 %i, 0.0, .3107 %i + .6036, 0.0, 1.811 %i - .1036] (%i5) L2 : inverse_fft (L1); (%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0, 4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0, - 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0] (%i6) lmax (abs (L2 - L)); (%o6) 3.545L-16
Complex data.
(%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $ (%i4) L1 : fft (L); (%o4) [0.5, .3536 %i + .3536, - 0.25 %i - 0.25, 0.5 - 6.776L-21 %i, 0.0, - .3536 %i - .3536, 0.25 %i - 0.25, 0.5 - 3.388L-20 %i] (%i5) L2 : inverse_fft (L1); (%o5) [1.0 - 4.066E-20 %i, 1.0 %i + 1.0, 1.0 - 1.0 %i, - 1.008L-19 %i - 1.0, 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 1.0 %i + 1.0, 1.947L-20 %i + 1.0] (%i6) lmax (abs (L2 - L)); (%o6) 6.83L-17
Computation of sine and cosine coefficients.
(%i1) load (fft) $ (%i2) fpprintprec : 4 $ (%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $ (%i4) n : length (L) $ (%i5) x : make_array (any, n) $ (%i6) fillarray (x, L) $ (%i7) y : fft (x) $ (%i8) a : make_array (any, n/2 + 1) $ (%i9) b : make_array (any, n/2 + 1) $ (%i10) a[0] : realpart (y[0]) $ (%i11) b[0] : 0 $ (%i12) for k : 1 thru n/2 - 1 do (a[k] : realpart (y[k] + y[n - k]), b[k] : imagpart (y[n - k] - y[k])); (%o12) done (%i13) a[n/2] : y[n/2] $ (%i14) b[n/2] : 0 $ (%i15) listarray (a); (%o15) [4.5, - 1.0, - 1.0, - 1.0, - 0.5] (%i16) listarray (b); (%o16) [0, - 2.414, - 1.0, - .4142, 0] (%i17) f(j) := sum (a[k] * cos (2*%pi*j*k / n) + b[k] * sin (2*%pi*j*k / n), k, 0, n/2) $ (%i18) makelist (float (f (j)), j, 0, n - 1); (%o18) [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]
Returns a rearranged representation of expr as in Horner's rule, using
x as the main variable if it is specified. x
may be omitted in
which case the main variable of the canonical rational expression form of
expr is used.
horner
sometimes improves stability if expr
is
to be numerically evaluated. It is also useful if Maxima is used to
generate programs to be run in Fortran. See also stringout
.
(%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155; 2 (%o1) 1.0E-155 x - 5.5 x + 5.2E+155 (%i2) expr2: horner (%, x), keepfloat: true; (%o2) (1.0E-155 x - 5.5) x + 5.2E+155 (%i3) ev (expr, x=1e155); Maxima encountered a Lisp error: floating point overflow Automatically continuing. To reenable the Lisp debugger set *debugger-hook* to nil. (%i4) ev (expr2, x=1e155); (%o4) 7.0E+154
Finds a root of the expression expr or the function f over the
closed interval [a, b]. The expression expr may be an
equation, in which case find_root
seeks a root of
lhs(expr) - rhs(expr)
.
Given that Maxima can evaluate expr or f over
[a, b] and that expr or f is continuous,
find_root
is guaranteed to find the root, or one of the roots if there
is more than one.
find_root
initially applies binary search.
If the function in question appears to be smooth enough,
find_root
applies linear interpolation instead.
The accuracy of find_root
is governed by find_root_abs
and
find_root_rel
. find_root
stops when the function in question
evaluates to something less than or equal to find_root_abs
,
or if successive approximants x_0, x_1 differ by no more than
find_root_rel * max(abs(x_0), abs(x_1))
. The default values of
find_root_abs
and find_root_rel
are both zero.
find_root
expects the function in question to have a different sign at
the endpoints of the search interval. When the function evaluates to a number
at both endpoints and these numbers have the same sign, the behavior of
find_root
is governed by find_root_error
.
When find_root_error
is true
, find_root
prints an error
message. Otherwise find_root
returns the value of
find_root_error
. The default value of find_root_error
is
true
.
If f evaluates to something other than a number at any step in the search
algorithm, find_root
returns a partially-evaluated find_root
expression.
The order of a and b is ignored; the region in which a root is sought is [min(a, b), max(a, b)].
Examples:
(%i1) f(x) := sin(x) - x/2; x (%o1) f(x) := sin(x) - - 2 (%i2) find_root (sin(x) - x/2, x, 0.1, %pi); (%o2) 1.895494267033981 (%i3) find_root (sin(x) = x/2, x, 0.1, %pi); (%o3) 1.895494267033981 (%i4) find_root (f(x), x, 0.1, %pi); (%o4) 1.895494267033981 (%i5) find_root (f, 0.1, %pi); (%o5) 1.895494267033981 (%i6) find_root (exp(x) = y, x, 0, 100); x (%o6) find_root(%e = y, x, 0.0, 100.0) (%i7) find_root (exp(x) = y, x, 0, 100), y = 10; (%o7) 2.302585092994046 (%i8) log (10.0); (%o8) 2.302585092994046
Returns an approximate solution of expr = 0
by Newton's method,
considering expr to be a function of one variable, x.
The search begins with x = x_0
and proceeds until abs(expr) < eps
(with expr evaluated at the current value of x).
newton
allows undefined variables to appear in expr,
so long as the termination test abs(expr) < eps
evaluates
to true
or false
.
Thus it is not necessary that expr evaluate to a number.
load(newton1)
loads this function.
See also realroots
, allroots
, find_root
, and
mnewton
.
Examples:
(%i1) load (newton1); (%o1) /usr/share/maxima/5.10.0cvs/share/numeric/newton1.mac (%i2) newton (cos (u), u, 1, 1/100); (%o2) 1.570675277161251 (%i3) ev (cos (u), u = %); (%o3) 1.2104963335033528E-4 (%i4) assume (a > 0); (%o4) [a > 0] (%i5) newton (x^2 - a^2, x, a/2, a^2/100); (%o5) 1.00030487804878 a (%i6) ev (x^2 - a^2, x = %); 2 (%o6) 6.098490481853958E-4 a
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The fourie
package comprises functions for the symbolic computation
of Fourier series.
There are functions in the fourie
package to calculate Fourier integral
coefficients and some functions for manipulation of expressions.
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Returns true
if equal (x, y)
otherwise false
(doesn't give an error message like equal (x, y)
would do in this case).
remfun (f, expr)
replaces all occurrences of
f (arg)
by arg in expr.
remfun (f, expr, x)
replaces all occurrences of
f (arg)
by arg in expr only if arg
contains the variable x.
funp (f, expr)
returns true
if expr contains the function f.
funp (f, expr, x)
returns true
if expr contains the function f and the variable
x is somewhere in the argument of one of the instances of f.
absint (f, x, halfplane)
returns the indefinite integral of f with respect to
x in the given halfplane (pos
, neg
, or both
).
f may contain expressions of the form
abs (x)
, abs (sin (x))
, abs (a) * exp (-abs (b) * abs (x))
.
absint (f, x)
is equivalent to
absint (f, x, pos)
.
absint (f, x, a, b)
returns the definite integral
of f with respect to x from a to b.
f may include absolute values.
Returns a list of the Fourier coefficients of f(x)
defined
on the interval [-p, p]
.
Simplifies sin (n %pi)
to 0 if sinnpiflag
is true
and
cos (n %pi)
to (-1)^n
if cosnpiflag
is true
.
Default value: true
See foursimp
.
Default value: true
See foursimp
.
Constructs and returns the Fourier series from the list of
Fourier coefficients l up through limit terms (limit
may be inf
). x and p have same meaning as in
fourier
.
Returns the Fourier cosine coefficients for f(x)
defined on
[0, p]
.
Returns the Fourier sine coefficients for f(x)
defined on
[0, p]
.
Returns fourexpand (foursimp (fourier (f, x, p)),
x, p, 'inf)
.
Constructs and returns a list of the Fourier integral coefficients of
f(x)
defined on [minf, inf]
.
Returns the Fourier cosine integral coefficients for f(x)
on
[0, inf]
.
Returns the Fourier sine integral coefficients for f(x)
on
[0, inf]
.
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