Periodic Diagonal Matrices
Periodic Diagonal Matrices
A
MN
Diagonal means that the matrix is of the form p
[
N ]
e dm cm bm a bp cp dp
(1.1)
4, 3, 2, 1, 0, 1, 2, 3
Or without e
A
a bp cp dp bm a bp cp cm bm a bp
(1.2)
dm cm bm a
This is made periodic by extending it to an 8×8 array of the form
a bp cp dp e dm cm bm bm a bp cp dp e dm cm cm bm a bp cp dp e dm
A p
dm cm bm a bp cp dp e
e dm cm bm a bp cp dp dp cp e dp dm e cm dm bm cm a bm bp a cp bp bp cp dp e dm cm bm a
(1.3)
This allows the matrix A p
to be defined as a periodic array with period 8 such that
A[12]=A[12-8]=A[4]
This extension defines matrix multiplications of the form
C
MN
MN
L
3
4
B A
ML LN
L
3
4
B
ML
L
4 N
3 (1.4)
To be specific
C
4,3
B
4, 4
A
4,3
B
4, 3
A
3,3
B
4, 2
B
A
4,1 1,3
B
A
4,2 2,3
B
A
4,3 3,3
A
2,3
B
4, 1
A
1,3
B bm
B cm
B dm
B e
B
4,0 dp
B
4,1 cp
B
4,2 bp
B
A a
4,3 3,3
B
4,0
A
0,3
(1.5)
According to (1.1) A
-4,3
= A[7] = A[7-8]=A[-1] =bm
A
-3,3
= A[6] = A[-2]=cm
A
-2,3
= A[5] = A[-3]=dm
A
-1,3
= A[4] = A[-4]=e
A
0,3
= A[3] = dp
A
1,3
= A[2] = cp
A
2,3
= A[1] = bp
A
3,3
= A[0] = a
Note that the order of N and L in (1.4) are important.
Defining the inverse of A[j-n]
The inverse is defined by
M n
M / 2
A
1
A
, ,
(2.1)
Assume that A
-1
, like A is a function of the coordinate differences, so that (2.1) becomes
M n
M / 2
n
m
(2.2)
Define the transform pair of A over M points as t i
iT
;
M f m
m
T
1
T
M m
M / 2
exp
j 2
f t m i
(2.3)
T
N
M i
M / 2
exp
j 2
f t m i
This implies that the M values of A in (2.3) extend as A[m+M]=A[m]. Then
T
M
2
2
M M n
M /2 i
M /2 a
1
exp
j 2
n
M
M
M /2 a
exp
j 2
k
n
M
Evaluate the sum over n first
T
M
2
2
M
M
M / 2 i
M / 2 a
1
exp
j 2
T
T
M
M
2
2
2
M
M
i
M / 2 a
M
M / 2 i
M
1
/ 2 a
1
exp
j 2
exp
j 2
k
M
For a
-1
[i]=1/(T
2
a[i]), this is satisfied. k
mi
M k
mi
M
M
n
M / 2 exp
j 2
n
M
, i
i
M
(2.4)
This is the equation used to test the inversion in MatInvFFT.doc
The –T/2 to T/2 range is treated in ..\..\Fourier\Symmetric range.doc
htm The zip
../../Fourier/for/symfft.zip
contains the relevant fft code. This code is modified here to have input from an unformatted file and output to an unformatted file.
O
PERATIONS REQUIRED
Using
j
T
1 N m
N / 2 a
1
T
N m
N / 2
Or exp
j 2
f t m j
1 exp
j 2
f t m j
(2.5)
(2.6)
A
1
1 M j
M / 2
1 exp
j 2
f t m j
exp
j 2
f t m j
(2.7)
M
N m
N / 2
This is an FFT from A to a, then a division and finally an FFT from a
-1
to A
-1
. The time required goes as 2Mlog(M)
T
ESTING
The specific test of interest is to generate a set of N values in the time domain. These are then transformed to N frequency values. The frequency values are truncated to M values.
This truncation leaves Time alone, but there are now only M points. The original time spacing of Time/N goes to
Time/M with fewer points. A transform over these fewer points is periodic in M, rather than N. for/DiagMat.wpj
diagmat2p.zip
Af linear
-0.108108E+00 0.229049E+01 0.000000E+00 The matrix of 16 values in time is truncated to 8 in frequency. Note
-0.810811E-01 0.276179E+01 -0.290837E-16 that the frequency does not go to zero at the ends
-0.540541E-01 0.317146E+01 0.641848E-16
-0.270270E-01 0.345038E+01 -0.691992E-16
0.000000E+00 0.354932E+01 0.000000E+00
0.270270E-01 0.345038E+01 -0.131378E-15
0.540541E-01 0.317146E+01 0.000000E+00
0.810811E-01 0.276179E+01 -0.912627E-16
Afinv(f)
-0.108108E+00 -0.199833E+01 -0.126535E-14 The inverse matrix is small indicating that more terms could have been
-0.810811E-01 0.106351E+01 0.728372E-15
-0.540541E-01 -0.153619E-01 -0.374458E-15
used. Definitely not diagonal.
-0.270270E-01 -0.186530E-01 0.279078E-15
0.000000E+00 -0.200229E-01 -0.256948E-15
0.270270E-01 -0.186530E-01 0.920926E-15
0.540541E-01 -0.153619E-01 -0.140141E-14
0.810811E-01 0.106351E+01 0.137022E-14
Afinv * AF
F -0.108E+00-0.811E-01-0.541E-01-0.270E-01 0.000E+00 0.270E-01 0.541E-01 0.811E-01
R 0.100E+01 0.749E-15-0.430E-15 0.312E-15-0.138E-14-0.104E-15-0.583E-15 0.319E-15
I 0.369E-15 0.109E-17 0.201E-15-0.484E-16-0.369E-15-0.109E-17-0.201E-15 0.484E-16
F -0.108E+00-0.811E-01-0.541E-01-0.270E-01 0.000E+00 0.270E-01 0.541E-01 0.811E-01
R 0.576E-15 0.100E+01 0.541E-15-0.430E-15 0.104E-15-0.139E-14 0.132E-15-0.749E-15
I 0.484E-16 0.369E-15 0.109E-17 0.201E-15-0.484E-16-0.369E-15-0.109E-17-0.201E-15
F -0.108E+00-0.811E-01-0.541E-01-0.270E-01 0.000E+00 0.270E-01 0.541E-01 0.811E-01
R -0.888E-15 0.444E-15 0.100E+01 0.444E-15-0.444E-15 0.000E+00-0.178E-14 0.000E+00
I -0.201E-15 0.484E-16 0.369E-15 0.109E-17 0.201E-15-0.484E-16-0.369E-15-0.109E-17
F -0.108E+00-0.811E-01-0.541E-01-0.270E-01 0.000E+00 0.270E-01 0.541E-01 0.811E-01
R -0.888E-15-0.888E-15 0.000E+00 0.100E+01 0.888E-15 0.000E+00 0.000E+00-0.178E-14
I -0.109E-17-0.201E-15 0.484E-16 0.369E-15 0.109E-17 0.201E-15-0.484E-16-0.369E-15
F -0.108E+00-0.811E-01-0.541E-01-0.270E-01 0.000E+00 0.270E-01 0.541E-01 0.811E-01
R -0.178E-14-0.444E-15-0.888E-15 0.444E-15 0.100E+01 0.444E-15-0.444E-15 0.000E+00
I -0.369E-15-0.109E-17-0.201E-15 0.484E-16 0.369E-15 0.109E-17 0.201E-15-0.484E-16
F -0.108E+00-0.811E-01-0.541E-01-0.270E-01 0.000E+00 0.270E-01 0.541E-01 0.811E-01
R 0.194E-15-0.154E-14 0.833E-16-0.687E-15 0.569E-15 0.100E+01 0.687E-15-0.576E-15
I -0.484E-16-0.369E-15-0.109E-17-0.201E-15 0.484E-16 0.369E-15 0.109E-17 0.201E-15
F -0.108E+00-0.811E-01-0.541E-01-0.270E-01 0.000E+00 0.270E-01 0.541E-01 0.811E-01
R -0.597E-15 0.298E-15-0.172E-14 0.298E-15-0.666E-15 0.756E-15 0.100E+01 0.805E-15
I 0.201E-15-0.484E-16-0.369E-15-0.109E-17-0.201E-15 0.484E-16 0.369E-15 0.109E-17
F -0.108E+00-0.811E-01-0.541E-01-0.270E-01 0.000E+00 0.270E-01 0.541E-01 0.811E-01
R 0.486E-15-0.527E-15 0.416E-16-0.173E-14 0.763E-16-0.680E-15 0.486E-15 0.100E+01
I 0.109E-17 0.201E-15-0.484E-16-0.369E-15-0.109E-17-0.201E-15 0.484E-16 0.369E-15
