thomas algorithm to solve tridiagonal matrices. basically sets up an lu decomposition three parts 1)...
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Thomas algorithm to solve tridiagonal matrices
rTx
nnnn
nnn
nnn
r
r
r
x
x
x
ba
cba
cba
cba
cba
cb
2
1
2
1
111
222
333
222
11
*
Basically sets up an LU decomposition
three parts
1) decomposition
2) forward substitution
3) backward substitution
1) decompositionloop from rows 2 to n
ai = ai/bi-1
bi = bi-ai*ci-1
end loop
2) forward substitutionloop from 2 to n
ri = ri - ai*ri-1
end loop
3) back substitutionxn = rn/bn
loop from n-1 to 1xi = (ri-ci*xi+1)/bi
end loop
Example
16
55
13
49
9
21
543
311
1214
25
5
4
3
2
1
x
x
x
x
x
First decompose T
loop from rows 2 to 5ai = ai/bi-1
bi = bi-ai*ci-1
end loop
21
543
311
1214
25
0515.11*0515.01
0515.0)4.19/(1
5/2 192*)5/4(21
5/4
3
3
2
2
b
a
b
a
0968.32194.0
55588.48529.2
30515.10515.0
14.198.0
25
forward substitutionloop from 2 to n
ri = ri - ai*ri-1
end loop
16
55
13
49
9
r
0968.32194.0
55588.48529.2
30515.10515.0
14.198.0
25
T
1546.158.41*0515.013
8.419*8.049
3
2
r
r
5806.18
7647.11
1546.15
8.41
9
r
back substitutionxn = rn/bn
loop from n-1 to 1xi = (ri-ci*xi+1)/bi
end loop
0968.32194.0
55588.48529.2
30515.10515.0
14.198.0
25
T
5806.18
7647.11
1546.15
8.41
9
r
15/)2*29(
24.19/)3*18.41(
30515.1/)4*)3(1546.15(
45588.4/)6*)5(7647.11(
60968.3/5806.18
1
2
3
4
5
x
x
x
x
x
6
4
3
2
1
x
Crout algorithm - alternate LU decomposition
Instead of 1’s on diagonal of L, get 1’s on diagonal of U
Operate on rows and columns sequentially, narrowing down to single element
Algorithm
n to1ifor 1,1, ii al
5554535251
4544434241
3534333231
2524232221
1514131211
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa
5554535251
4544434241
3534333231
2524232221
1514131211
aaaal
aaaal
aaaal
aaaal
aaaal
5554535251
4544434241
3534333231
2524232221
1514131211
aaaal
aaaal
aaaal
aaaal
uuuul
for j=2 to n-1
n to1jkfor
1
1,
jj
j
iikjijk
kj l
ulau
n tojifor 1
1,,
ul-alj
kkjikjiji
5554535251
4544434241
3534333231
2524232221
1514131211
aaall
aaall
aaall
aaall
uuuul
5554535251
4544434241
3534333231
2524232221
1514131211
aaall
aaall
aaall
uuull
uuuul
5554535251
4544434241
3534333231
2524232221
1514131211
aalll
aalll
aalll
uuull
uuuul
n to2jfor 11
11
l
au j
j
5554535251
4544434241
3534333231
2524232221
1514131211
aalll
aalll
uulll
uuull
uuuul
5554535251
4544434241
3534333231
2524232221
1514131211
allll
allll
uulll
uuull
uuuul
For j=n
1
1
n
kknnknnnn ulal
5554535251
4544434241
3534333231
2524232221
1514131211
allll
ullll
uulll
uuull
uuuul
5554535251
4544434241
3534333231
2524232221
1514131211
lllll
ullll
uulll
uuull
uuuul
Cholesky decomposition
A decomposition for symmetric matrices
Symmetric matrix e.g. a covariance matrix
332313
322212
312111
covcovcov
covcovcov
covcovcov
xxxxxx
xxxxxx
xxxxxx
For symmetric matrices, can write
T
T
LU
LLA
i.e.
Can develop relationships for l
1
1
2
1
1 1-k to1ifor
k
j
kjkkkk
ii
i
jkjijki
ki
lal
l
lla
l
Example:
135.025.015.0
35.013.04.0
25.04.012.0
15.04.02.01
A
1st row
11
1
12
1111
1
111
1 1-1 to1ifor
j
j
ii
i
jjiji
i
lal
l
lla
lSkip this equation
1111 al
135.025.015.0
35.013.04.0
25.04.012.0
15.04.02.01
A
1
L
Row 2
1
1
22
2222
0
122
2 1ifor
j
j
ii
jjiji
i
lal
l
lla
l
9798.02.01
1
2.0
221
22222
11
2121
lal
l
al
135.025.015.0
35.013.04.0
25.04.012.0
15.04.02.01
A
9798.02.0
1
L
Row 3
8886.02245.04.01
2245.09798.0
4.0*2.03.0
1)(i 1
4.0
2232
231
23333
22
31213232
11
3131
llal
l
llal
l
al
135.025.015.0
35.013.04.0
25.04.012.0
15.04.02.01
A
8886.02245.04.0
9798.02.0
1
L
9243.02696.02245.015.01
2696.08886.0
2245.0*2245.015.0*4.035.0
2245.09798.0
15.0*2.025.0
1
15.0
22243
242
241
24444
33
423241314343
22
41214242
11
4141
lllal
l
llllal
l
llal
l
al
Row 4
135.025.015.0
35.013.04.0
25.04.012.0
15.04.02.01
A
9243.02696.02245.015.0
8886.02245.04.0
9798.02.0
1
L
9243.0
2696.08886.0
2245.02245.09798.0
15.04.02.01
TL
Iterative methods for solving matrix equations
1. Jacobi
2. Gauss-Seidel
3. Successive overrelaxation (SOR)
Idea behind iterative methods: Convert Ax=b into Cx=d, which has sequence of approximations x(1), x(2), … with
dCxx kk 1
What are C and d?
Recall from fixed-point iteration
xxxx
xxxx
sin0sin2
3032
22
Rewrite matrix equation in same way
4444343242141
3434333232131
2424323222121
1414313212111
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
becomes
44
43
44
432
44
421
44
414
33
34
33
342
33
321
33
313
22
24
22
243
22
231
22
212
11
14
11
143
11
132
11
121
a
bx
a
ax
a
ax
a
ax
a
bx
a
ax
a
ax
a
ax
a
bx
a
ax
a
ax
a
ax
a
bx
a
ax
a
ax
a
ax
Then
0
0
0
0
44
43
44
42
44
41
33
34
33
32
33
31
22
24
22
23
22
21
11
14
11
13
11
12
a
a
a
a
a
aa
a
a
a
a
aa
a
a
a
a
aa
a
a
a
a
a
C
44
4
33
3
22
2
11
1
a
ba
ba
ba
b
d
Jacobi method is like fixed point iteration
Example: Shape of a stretched membrane
Shape can be described by potential function
0 yyxx uu
Let us give some boundary conditions
11,
0,
1,1
,0
10 ,10 0
2
2
xu
xxu
yu
yyu
yxuu yyxx
1111111
164.08.0
136.06.0
116.04.0
104.02.0
164.036.016.004.00
18.06.04.02.0
16151413
1211109
8765
4321
uuuu
uuuu
uuuu
uuuu
y
x
Problem look likes this
Solve for u’s
2
1
1
64.1
1
0
0
36.0
1
0
0
16.0
64.1
36.0
16.0
08.0
411
1411
1411
141
1411
11411
11411
1141
1411
11411
11411
1141
141
1141
1141
114
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
Leads to this system of equations
Choose an initial u=[1 1 1 1 …1]’
Iterate using x=Cx+d
Matlab solution, 49 iterations
Gauss-Seidel method differs from Jacobi by sequential updating
- use new xi’s as they become available
Example:
154
62
13223
321
321
321
xxx
xxx
xxx
4
15
4
1
4
1
32
1
2
13
13
3
2
3
2
213
312
321
oldoldnew
oldoldnew
oldoldnew
xxx
xxx
xxx
4
15
4
1
4
1
32
1
2
13
13
3
2
3
2
213
312
321
newnewnew
oldnewnew
oldoldnew
xxx
xxx
xxx
Jacobi Gauss-Seidel
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