The treatment of large structural systems may be simplified by dividing the system into smaller systems called components. The components are related through the displacement, and force conditions at their junction points. Each component is represented by mode shapes (or functions).

1. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 1 of 11
Component Mode Synthesis (CMS) Analysis
by Julio C. Banks
The treatment of large structural systems may be simplified by dividing the system into
smaller systems called components. The components are related through the
displacement, and force conditions at their junction points. Each component is represented
by mode shapes (or functions). The sum of the component mode shape functions allows
the satisfaction of the displacement and force conditions at the junctions [1].
≔
α 1 ≔
m 1 ＝
α ―
L2
L1
≔
L1 1 ≔
L2 ⋅
α L1
Component 1: ≔
M
,
1 1
―
1
5
≔
M
,
1 2
―
1
6
≔
M
,
2 1
M
,
1 2
≔
M
,
2 2
―
1
7
Component 2: ≔
M
,
3 3
⋅
1.0 α ≔
M
,
3 4
⋅
―
1
2
α ≔
M
,
3 5
⋅
―
1
5
α
≔
M
,
4 3
M
,
3 4
≔
M
,
4 4
⋅
―
1
3
α ≔
M
,
4 5
⋅
―
1
6
α
≔
M
,
5 3
M
,
3 5
≔
M
,
5 4
M
,
4 5
≔
M
,
5 5
⋅
―
1
9
α
Component 3: ≔
M
,
6 6
α
Julio C. Banks

2. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 2 of 11
≔
M
,
6 6
α
=
M
0.2000 0.1667 0.0000 0.0000 0.0000 0.0000
0.1667 0.1429 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 1.0000 0.5000 0.2000 0.0000
0.0000 0.0000 0.5000 0.3333 0.1667 0.0000
0.0000 0.0000 0.2000 0.1667 0.1111 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⋅
m L1
≔
E 1 ≔
I 1 ≔
K
,
1 1
4 ≔
K
,
1 2
6 ≔
K
,
2 1
K
,
1 2
≔
K
,
2 2
12 ≔
K
,
5 5
⋅
28.8 ―
1
α
3
≔
K
,
6 6
0
=
K
4.00 6.00 0.00 0.00 0.00 0.00
6.00 12.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 28.80 0.00
0.00 0.00 0.00 0.00 0.00 0.00
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
――
⋅
E I
L1
3
＝
⋅
1 1 0 0 0 1
0 0 1 1 1 0
2 3 0 -―
1
α
-―
4
α
0
2 6 0 0 ―
12
α
2
0
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
p1
p2
p3
p4
P5
P6
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
0
Apply the nonzero diagonal criterion to select the independent (generalized) coordinates, p.
＝
⋅
1 0 0 0
0 1 1 1
2 0 -―
1
α
-―
4
α
2 0 0 ―
12
α
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
p1
p3
p4
P5
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
⋅
-
1 1
0 0
3 0
6 0
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
p2
p6
⎡
⎢
⎣
⎤
⎥
⎦
Julio C. Banks

3. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 3 of 11
Or ＝
⋅
S p ⋅
Q q
where ＝
S
1 0 0 0
0 1 1 1
2 0 -―
1
α
-―
4
α
2 0 0 ―
12
α
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
and ＝
Q
-1 -1
0 0
-3 0
-6 0
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
Let ＝
p2 q1 and ＝
p6 q2 therefore,
＝
⋅
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 1 1 0
2 0 0 -―
1
α
-―
4
α
0
2 0 0 0 ―
12
α
2
0
0 0 0 0 0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
p1
p2
p3
p4
P5
P6
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⋅
-1 -1
1 0
0 0
-3 0
-6 0
0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
q1
q2
⎡
⎢
⎣
⎤
⎥
⎦
or ＝
⋅
S' p ⋅
Q' q
where ≔
S'
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 1 1 0
2 0 0 -―
1
α
-―
4
α
0
2 0 0 0 ―
12
α
2
0
0 0 0 0 0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
and ≔
Q'
-1 -1
1 0
0 0
-3 0
-6 0
0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Therefore, ＝
p ⋅
T q Where ≔
T =
⋅
(
(S')
)-1
Q'
-1.000 -1.000
1.000 0.000
-2.000 2.500
2.333 -2.667
-0.333 0.167
0.000 1.000
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Julio C. Banks

4. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 4 of 11
≔
A =
⋅
⋅
T
T M T
1.18 -1.48
-1.48 3.18
⎡
⎢
⎣
⎤
⎥
⎦
≔
B =
⋅
⋅
T
T K T
7.200 -3.600
-3.600 4.800
⎡
⎢
⎣
⎤
⎥
⎦
≔
C =
⋅
A
-1
B
11.40 -2.80
4.19 0.20
⎡
⎢
⎣
⎤
⎥
⎦
≔
λ =
sort(
(eigenvals(
(C)
))
)
1.37
10.23
⎡
⎢
⎣
⎤
⎥
⎦
≔
ω =
→
―
‾‾
λ
1.172
3.198
⎡
⎢
⎣
⎤
⎥
⎦
‾‾‾‾‾‾
―――
⋅
E I
⋅
m L1
4
Calculate the eigenvectors ≔
Nm =
length(
(λ)
) 2
≔
i ‥
1 Nm
≔
Φ
⟨
⟨i⟩
⟩
eigenvec⎛
⎝
,
C λ
i
⎞
⎠
The mode shapes in normal coordinates is =
Φ
0.2693 -0.9225
0.9631 -0.3859
⎡
⎢
⎣
⎤
⎥
⎦
Normalized mode shapes ≔
Φn =
Vnorm
(
(Φ)
)
0.280 1.000
1.000 0.418
⎡
⎢
⎣
⎤
⎥
⎦
First and Second Mode Shapes
Reference
Ÿ
Ÿ
"Theory of Vibration with Applications, 5th Ed.", Thomson, W. T., and Marie Dillon
Dahleh. Prentice Hall. ISBN 0-13-651068-X, Pp. 341 through 346.
Julio C. Banks MSME Thesis - "Component Synthesis Methods for Vibrating Systems".
Tufts University, Medford Massachusetts, May 1984.
Appendix A
Julio C. Banks

5. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 5 of 11
Appendix A
＝
⋅
1 1 0 0 0 1
0 0 1 1 1 0
2 3 0 -1 -4 0
2 6 0 0 12 0
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
p1
p2
p3
p4
P5
P6
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
0
Since the total number of coordinates used are six and there are four constraint equations,
the number of generalized coordinates for the system is two (i.e., there are four
superfluous coordinates corresponding to the four constraint equations. We can thus
choose any two (the first example uses the nonzero diagonal criterion) of the generalized
coordinates, q. Let p1 = q1, and p6 = q6 be the generalized coordinates, and express
p1 ..p6 in terms of q1, and q6 according to the following steps:
＝
⋅
1 0 0 0
0 1 1 1
3 0 -―
1
α
-―
4
α
6 0 0 ―
12
α
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
p1
p3
p4
P5
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
⋅
-
1 1
0 0
2 0
2 0
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
q2
q6
⎡
⎢
⎣
⎤
⎥
⎦
or ＝
⋅
S p ⋅
Q q
Let ＝
p1 q1
where ＝
S
1 0 0 0
0 1 1 1
3 0 -―
1
α
-―
4
α
6 0 0 ―
12
α
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
and ＝
Q
-1 -1
0 0
-2 0
-2 0
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦ and ＝
p6 q2
＝
⋅
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 1 1 0
3 0 0 -―
1
α
-―
4
α
0
6 0 0 0 ―
12
α
2
0
0 0 0 0 0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
p1
p2
p3
p4
P5
P6
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⋅
1 0
-1 -1
0 0
-2 0
-2 0
0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
q1
q2
⎡
⎢
⎣
⎤
⎥
⎦
or ＝
⋅
S' p ⋅
Q' q
Julio C. Banks

6. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 6 of 11
For =
α 1.00
Where ≔
S'
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 1 1 0
0 3 0 -―
1
α
-―
4
α
0
0 6 0 0 ―
12
α
2
0
0 0 0 0 0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
and ≔
Q'
1 0
-1 -1
0 0
-2 0
-2 0
0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
＝
p ⋅
T. q Where ≔
T =
⋅
(
(S')
)-1
Q'
1.00 0.00
-1.00 -1.00
2.00 4.50
-2.33 -5.00
0.33 0.50
0.00 1.00
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
≔
A =
⋅
⋅
T
T M T
1.1774 2.6614
2.6614 7.3206
⎡
⎢
⎣
⎤
⎥
⎦
≔
B =
⋅
⋅
T
T K T
7.200 10.800
10.800 19.200
⎡
⎢
⎣
⎤
⎥
⎦
≔
C =
⋅
A
-1
B
15.60 18.20
-4.19 -3.99
⎡
⎢
⎣
⎤
⎥
⎦
≔
λ =
sort(
(eigenvals(
(C)
))
)
1.37
10.23
⎡
⎢
⎣
⎤
⎥
⎦
≔
ω =
→
―
‾‾
λ
1.172
3.198
⎡
⎢
⎣
⎤
⎥
⎦
‾‾‾‾‾‾
―――
⋅
E I
⋅
m L1
4
Example
Julio C. Banks

7. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 7 of 11
Example
≡
gc ⋅
g ――
lb
lbf
≔
γ ⋅
0.280 ――
lbf
in
3
＝
γ ⋅
ρ ―
g
gc
⇒ ≔
ρ =
⋅
γ ―
gc
g
0.280 ――
lb
in
3
≔
Do ⋅
1.315 in ≔
Di ⋅
1.049 in ≔
L1 ⋅
10 in ≔
E ⋅
⋅
30 10
6
psi
Area: ≔
A =
⋅
―
π
4
⎛
⎝ -
Do
2
Di
2 ⎞
⎠ 0.4939 in
2
Moment of Inertia: ≔
I =
⋅
―
π
64
⎛
⎝ -
Do
4
Di
4 ⎞
⎠
⎛
⎝ ⋅
8.734 10
-2⎞
⎠ in
4
Thickness: ≔
t =
―――
-
Do Di
2
0.003 m
≔
V =
⋅
A L1 4.94 in
3
≔
m =
⋅
ρ A 0.1383 ―
lb
in
≔
ω' =
⋅
ω
⎛
⎜
⎜
⎝
‾‾‾‾‾‾
―――
⋅
E I
⋅
m L1
4
⎞
⎟
⎟
⎠
1.172
⋮
⎡
⎢
⎣
⎤
⎥
⎦
‾‾‾‾‾‾
―――
⋅
E I
⋅
m L1
4
≔
f =
――
ω'
⋅
2 π
159.6
⋮
⎡
⎢
⎣
⎤
⎥
⎦
Hz
FEM Validation:
ANSYS FEA ≔
f' ⋅
159.2
433.0
⎡
⎢
⎣
⎤
⎥
⎦
Hz ≔
error =
→
―
――
-
f' f
f
-0.24
-0.55
⎡
⎢
⎣
⎤
⎥
⎦
%
1
CAEFEM FEA ≔
f' ⋅
165
470
⎡
⎢
⎣
⎤
⎥
⎦
Hz ≔
error =
→
―
――
-
f' f
f
3.4
8.0
⎡
⎢
⎣
⎤
⎥
⎦
%
1
Results Commentary
ANSYS model uses 2D Beam Elements, while CAEFEM model utilized 3D Beam Elements
(That is, ANSYS will most closely follow the closed-form solution since the latter is 2D. On
the other hand, CAEFEM model had to be restrained in the out-of-plane dimension in order
to emulate a 2D plane frame (CAEFEM has 3D beam elements only).
Julio C. Banks

8. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 8 of 11
Results Commentary
ANSYS model uses 2D Beam Elements, while CAEFEM model utilized 3D Beam Elements
(That is, ANSYS will most closely follow the closed-form solution since the latter is 2D. On
the other hand, CAEFEM model had to be restrained in the out-of-plane dimension in order
to emulate a 2D plane frame (CAEFEM has 3D beam elements only).
In general, the natural frequencies can be expressed as a function of . The choices
＝
α ―
L2
L1
of dependent, and independent coordinates follows those chosen in the reference. The
results are identical.
The mass matrix: The Stiffness matrix:
≔
M(
(α)
)
―
1
5
―
1
6
0 0 0 0
―
1
6
―
1
7
0 0 0 0
0 0 α ⋅
―
1
2
α ⋅
―
1
5
α 0
0 0 ⋅
―
1
2
α ⋅
―
1
3
α ⋅
―
1
6
α 0
0 0 ⋅
―
1
5
α ⋅
―
1
6
α ⋅
―
1
9
α 0
0 0 0 0 0 α
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
≔
K(
(α)
)
4 6 0 0 0 0
6 12 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 ――
28.8
α
3
0
0 0 0 0 0 0
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Recall ≔
S'(
(α)
)
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 1 1 0
2 0 0 -―
1
α
-―
4
α
0
2 0 0 0 ―
12
α
2
0
0 0 0 0 0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
and ≔
Q'
-1 -1
1 0
0 0
-3 0
-6 0
0 1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
＝
p ⋅
T q Where ≔
T(
(α)
) ⋅
(
(S'(
(α)
))
)-1
Q'
=
T(
(α)
)
-1.000 -1.000
1.000 0.000
-2.000 2.500
2.333 -2.667
-0.333 0.167
0.000 1.000
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
≔
α 1
Julio C. Banks

9. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 9 of 11
≔
α 1
≔
A(
(α)
) ⋅
⋅
T
T(
(α)
) M(
(α)
) T(
(α)
) ≔
B(
(α)
) ⋅
⋅
T
T(
(α)
) K(
(α)
) T(
(α)
)
=
A(
(α)
)
1.1774 -1.4840
-1.4840 3.1753
⎡
⎢
⎣
⎤
⎥
⎦
=
B(
(α)
)
7.2000 -3.6000
-3.6000 4.8000
⎡
⎢
⎣
⎤
⎥
⎦
≔
C(
(α)
) ⋅
A(
(α)
)
-1
B(
(α)
)
=
C(
(α)
)
11.40 -2.80
4.19 0.20
⎡
⎢
⎣
⎤
⎥
⎦
≔
λ =
sort(
(eigenvals(
(C(
(α)
))
))
)
1.37
10.23
⎡
⎢
⎣
⎤
⎥
⎦
≔
ω =
⋅
⎛
⎝
→
―
‾‾
λ
⎞
⎠
‾‾‾‾‾‾
―――
⋅
E I
⋅
m L1
4
1.172
⋮
⎡
⎢
⎣
⎤
⎥
⎦
‾‾‾‾‾‾
―――
⋅
E I
⋅
m L1
4
≔
f =
――
ω
⋅
2 π
159.6
435.4
⎡
⎢
⎣
⎤
⎥
⎦
Hz
Calculate the eigenvectors
≔
Nm =
length(
(λ)
) 2
≔
i ‥
1 Nm
≔
Φ
⟨
⟨i⟩
⟩
eigenvec⎛
⎝
,
C(
(α)
) λ
i
⎞
⎠
The mode shapes in normal coordinates is =
Φ
0.2693 -0.9225
0.9631 -0.3859
⎡
⎢
⎣
⎤
⎥
⎦
≔
D =
⋅
T(
(α)
) Φ
-1.232 1.308
0.269 -0.923
1.869 0.880
-1.940 -1.123
⎡
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
Julio C. Banks

10. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 10 of 11
The mode shapes in physical coordinates is ≔
D =
⋅
T(
(α)
) Φ
-1.232 1.308
0.269 -0.923
1.869 0.880
-1.940 -1.123
0.071 0.243
0.963 -0.386
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Normalized mode shapes ≔
Dn =
Vnorm
(
(D)
)
0.635 1.000
-0.139 -0.705
-0.964 0.673
1.000 -0.859
-0.036 0.186
-0.496 -0.295
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Julio C. Banks

11. Mathcad - CMS (Component Mode Synthesis) Analysis.mcdx Page 11 of 11
Appendix B
Define two (2) procedures to unit-normalize the columns of a matrix. The first algorithm
determines the maximum magnitude in each column. The second algorithm, expands the
first algorithm to the unit-normalization phase of the solution
≡
Vmax
(
(v)
) ⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
“Determine the maximum-magnitude”
“element in each column of a matrix”
←
Nr rows(
(v)
)
←
Nc cols(
(v)
)
for ∊ |
|
|
|
|
|
|
|
|
|
|
|
|
j ‥
1 Nc
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
←
max v
,
1 j
for ∊ |
|
|
|
|
|
|
i ‥
2 Nr
‖
‖
‖
‖
‖
|
|
|
|
|
if >
|
|
v
,
i j
|
|
|
|max|
|
‖
‖
‖
←
max v
,
i j
←
z
j
max
return z
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
≡
Vnorm
(
(v)
) ⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
“Determine the maximum-magnitude”
“element in each column of a matrix”
←
Nr rows(
(v)
)
←
Nc cols(
(v)
)
for ∊ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
j ‥
1 Nc
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
‖
←
max v
,
1 j
for ∊ |
|
|
|
|
|
|
i ‥
2 Nr
‖
‖
‖
‖
‖
|
|
|
|
|
if >
|
|
v
,
i j
|
|
|
|max|
|
‖
‖
‖
←
max v
,
i j
←
z
j
max
←
Vn
⟨
⟨j⟩
⟩
―
v
⟨
⟨j⟩
⟩
z
j
return Vn
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
Julio C. Banks