WolframAlpha.com
WolframCloud.com
All Sites & Public Resources...
Products & Services
Wolfram|One
Mathematica
Wolfram|Alpha Notebook Edition
Finance Platform
System Modeler
Wolfram Player
Wolfram Engine
WolframScript
Enterprise Private Cloud
Application Server
Enterprise Mathematica
Wolfram|Alpha Appliance
Enterprise Solutions
Corporate Consulting
Technical Consulting
Wolfram|Alpha Business Solutions
Resource System
Data Repository
Neural Net Repository
Function Repository
Wolfram|Alpha
Wolfram|Alpha Pro
Problem Generator
API
Data Drop
Products for Education
Mobile Apps
Wolfram Player
Wolfram Cloud App
Wolfram|Alpha for Mobile
Wolfram|Alpha-Powered Apps
Services
Paid Project Support
Wolfram U
Summer Programs
All Products & Services »
Technologies
Wolfram Language
Revolutionary knowledge-based programming language.
Wolfram Cloud
Central infrastructure for Wolfram's cloud products & services.
Wolfram Science
Technology-enabling science of the computational universe.
Wolfram Notebooks
The preeminent environment for any technical workflows.
Wolfram Engine
Software engine implementing the Wolfram Language.
Wolfram Natural Language Understanding System
Knowledge-based broadly deployed natural language.
Wolfram Data Framework
Semantic framework for real-world data.
Wolfram Universal Deployment System
Instant deployment across cloud, desktop, mobile, and more.
Wolfram Knowledgebase
Curated computable knowledge powering Wolfram|Alpha.
All Technologies »
Solutions
Engineering, R&D
Aerospace & Defense
Chemical Engineering
Control Systems
Electrical Engineering
Image Processing
Industrial Engineering
Mechanical Engineering
Operations Research
More...
Finance, Statistics & Business Analysis
Actuarial Sciences
Bioinformatics
Data Science
Econometrics
Financial Risk Management
Statistics
More...
Education
All Solutions for Education
Trends
Machine Learning
Multiparadigm Data Science
Internet of Things
High-Performance Computing
Hackathons
Software & Web
Software Development
Authoring & Publishing
Interface Development
Web Development
Sciences
Astronomy
Biology
Chemistry
More...
All Solutions »
Learning & Support
Learning
Wolfram Language Documentation
Fast Introduction for Programmers
Wolfram U
Videos & Screencasts
Wolfram Language Introductory Book
Webinars & Training
Summer Programs
Books
Need Help?
Support FAQ
Wolfram Community
Contact Support
Premium Support
Paid Project Support
Technical Consulting
All Learning & Support »
Company
About
Company Background
Wolfram Blog
Events
Contact Us
Work with Us
Careers at Wolfram
Internships
Other Wolfram Language Jobs
Initiatives
Wolfram Foundation
MathWorld
Computer-Based Math
A New Kind of Science
Wolfram Technology for Hackathons
Student Ambassador Program
Wolfram for Startups
Demonstrations Project
Wolfram Innovator Awards
Wolfram + Raspberry Pi
Summer Programs
More...
All Company »
Search
WOLFRAM COMMUNITY
Connect with users of Wolfram technologies to learn, solve problems and share ideas
Join
Sign In
Dashboard
Groups
People
Message Boards
Answer
(
Unmark
)
Mark as an Answer
GROUPS:
Staff Picks
Mathematics
Calculus
Equation Solving
Graphics and Visualization
Wolfram Language
Optimization
Modeling
Numerical Computation
5
Rauan Kaldybaev
Spherical harmonics interpolation method that minimizes an energy function
Rauan Kaldybaev
Posted
1 year ago
1976 Views
|
1 Reply
|
5 Total Likes
Follow this post
|
Spherical harmonics interpolation method that minimizes an energy function
by
Rauan Kaldybaev
Interpolation on a sphere is an essential problem in engineering, computer graphics, acoustics, and other fields, not to mention its mathematical interest. One approach to spherical interpolation is through spherical harmonics - these functions are smooth, numerically convenient, and easy to analyze. One issue, however, is that interpolation with spherical harmonics generally produces and overdetermined system, as spherical harmonics of degree
l
form a “package” of size
2
l
+
1
and can only meaningfully be viewed together as a basis for the space of homogeneous polynomials of degree
l
that satisfy Laplace’s equations. This essay presents a method of interpolation that resolves this overdetermination by using the excess degrees of freedom to minimize a given energy function. A wisely chosen energy function makes high-degree modes costly, thus discouraging extreme variation, and is zero for the constant function. The resulting interpolation evaluates exactly to the reference values. The root-mean-squared error was found to be very accurately (
2
r
=
0
.
9
8
4
) described by the empirical law
1
/
4
/
3
m
, where
m
is the number of data points. The interpolation respects spherical symmetry, so it does not introduce coordinate bias. To assess the effectiveness of min-energy interpolation, it was compared against the more traditional regularization-based interpolation on the real-world example of illumination of a sphere by lamps. With 110 data points, min-energy interpolation was found to be roughly 4 times more accurate than regularization-based interpolation. Then, to test the speed of convergence, the error of the two interpolations was plotted for different values of
m
; min-energy interpolation again performed better. The computation time of both interpolation methods is approximately the same. Min-energy interpolation is great for interpolation of expensive-to-compute mathematical functions. However, because it evaluates exactly to the reference values, it does not work well with noisy data. Min-energy interpolation basically corresponds to the limit of Tikhonov regularization with
λ
-
>
0
and a properly chosen regularization matrix.
O
u
t
[
]
=
Qualitative introduction. An overdetermined model
O
n
e
p
r
o
b
l
e
m
w
i
t
h
u
s
i
n
g
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
f
o
r
i
n
t
e
r
p
o
l
a
t
i
o
n
i
s
t
h
a
t
t
h
e
y
c
o
m
e
i
n
“
p
a
c
k
a
g
e
s
”
.
T
h
e
r
e
a
r
e
2
(
L
+
1
)
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
w
i
t
h
l
<
=
L
,
m
e
a
n
i
n
g
t
h
a
t
t
h
e
n
u
m
b
e
r
o
f
b
a
s
i
s
f
u
n
c
t
i
o
n
s
i
n
s
u
c
h
i
n
t
e
r
p
o
l
a
t
i
o
n
c
a
n
o
n
l
y
b
e
a
p
u
r
e
s
q
u
a
r
e
-
1
,
4
,
9
,
1
6
,
2
5
,
a
n
d
s
o
o
n
.
I
f
w
e
h
a
v
e
1
1
p
o
i
n
t
s
t
o
i
n
t
e
r
p
o
l
a
t
e
,
w
e
h
a
v
e
t
o
s
t
i
c
k
w
i
t
h
1
6
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
,
w
h
i
c
h
l
e
a
v
e
s
5
u
n
u
s
e
d
d
e
g
r
e
e
s
o
f
f
r
e
e
d
o
m
.
A
n
d
t
h
i
s
i
s
w
h
a
t
m
a
k
e
s
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
i
n
t
e
r
p
o
l
a
t
i
o
n
s
o
m
e
w
h
a
t
n
o
n
t
r
i
v
i
a
l
.
O
n
e
p
o
s
s
i
b
l
e
t
h
i
n
g
t
o
d
o
w
o
u
l
d
b
e
t
o
“
t
h
r
o
w
a
w
a
y
”
t
h
e
e
x
t
r
a
h
a
r
m
o
n
i
c
s
-
t
h
a
t
i
s
,
j
u
s
t
s
e
t
s
o
m
e
o
f
t
h
e
c
o
e
f
f
i
c
i
e
n
t
s
t
o
z
e
r
o
.
B
u
t
t
h
a
t
w
o
u
l
d
i
n
t
r
o
d
u
c
e
c
o
o
r
d
i
n
a
t
e
b
i
a
s
.
T
h
e
s
e
t
o
f
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
w
i
t
h
a
p
a
r
t
i
c
u
l
a
r
v
a
l
u
e
o
f
l
f
o
r
m
a
b
a
s
i
s
f
o
r
t
h
e
s
p
a
c
e
o
f
e
i
g
e
n
f
u
n
c
t
i
o
n
s
o
f
t
h
e
L
a
p
l
a
c
i
a
n
w
i
t
h
e
i
g
e
n
v
a
l
u
e
-
l
(
l
+
1
)
.
I
f
w
e
f
o
r
c
e
f
u
l
l
y
r
e
m
o
v
e
s
o
m
e
o
f
t
h
e
h
a
r
m
o
n
i
c
s
,
w
e
w
o
u
l
d
b
e
i
m
p
o
s
i
n
g
a
n
a
r
t
i
f
i
c
i
a
l
c
o
n
s
t
r
a
i
n
t
t
h
a
t
c
o
n
t
r
a
d
i
c
t
s
t
h
e
s
y
m
m
e
t
r
y
o
f
t
h
e
p
r
o
b
l
e
m
.
A
n
d
t
h
i
s
,
i
n
t
u
r
n
,
c
a
n
d
r
a
m
a
t
i
c
a
l
l
y
d
e
c
r
e
a
s
e
a
c
c
u
r
a
c
y
.
A
s
a
n
e
x
a
m
p
l
e
,
i
m
a
g
i
n
e
t
h
a
t
w
e
a
r
e
a
s
k
e
d
t
o
f
i
t
t
h
e
l
i
n
e
a
r
m
o
d
e
l
ψ
[
x
,
y
]
=
ψ
0
+
v
x
x
+
v
y
y
t
o
t
h
e
d
a
t
a
ψ
[
1
,
3
]
=
0
,
ψ
[
6
,
1
5
]
=
2
6
.
T
h
e
r
e
a
r
e
3
d
e
g
r
e
e
s
o
f
f
r
e
e
d
o
m
a
n
d
o
n
l
y
2
c
o
n
s
t
r
a
i
n
t
s
.
O
n
e
o
p
t
i
o
n
w
o
u
l
d
b
e
t
o
j
u
s
t
s
e
t
v
y
=
0
,
w
h
i
c
h
l
e
a
d
s
t
o
v
x
=
2
.
6
.
H
o
w
e
v
e
r
,
t
h
i
s
w
o
u
l
d
m
e
a
n
c
h
o
o
s
i
n
g
t
h
e
x
-
a
x
i
s
a
s
t
h
e
p
r
e
f
e
r
r
e
d
d
i
r
e
c
t
i
o
n
,
a
n
d
h
a
v
i
n
g
p
r
e
f
e
r
r
e
d
d
i
r
e
c
t
i
o
n
s
j
u
s
t
d
o
e
s
n
’
t
f
e
e
l
r
i
g
h
t
.
F
r
o
m
a
m
o
r
e
p
r
a
c
t
i
c
a
l
p
e
r
s
p
e
c
t
i
v
e
,
t
h
e
v
a
l
u
e
v
x
=
5
.
2
i
s
r
a
t
h
e
r
e
x
t
r
e
m
e
:
a
m
o
r
e
r
e
a
l
i
s
t
i
c
e
s
t
i
m
a
t
e
f
o
r
t
h
e
n
o
r
m
o
f
v
w
o
u
l
d
b
e
2
6
2
(
6
-
1
)
+
2
(
1
5
-
3
)
=
2
.
T
h
e
o
n
l
y
r
i
g
h
t
d
i
r
e
c
t
i
o
n
f
o
r
v
i
s
f
r
o
m
(
1
,
3
)
t
o
(
6
,
1
5
)
.
A
n
d
h
o
w
e
x
a
c
t
l
y
d
o
w
e
f
i
n
d
“
t
h
e
r
i
g
h
t
d
i
r
e
c
t
i
o
n
”
i
n
a
m
o
r
e
g
e
n
e
r
a
l
c
a
s
e
?
F
o
r
m
p
o
i
n
t
s
o
n
a
s
p
h
e
r
e
,
t
h
e
s
e
t
o
f
c
o
e
f
f
i
c
i
e
n
t
s
i
s
n
’
t
e
v
e
n
a
v
e
c
t
o
r
(
i
t
’
s
m
o
r
e
s
i
m
i
l
a
r
t
o
a
m
u
l
t
i
-
i
n
d
e
x
t
e
n
s
o
r
)
.
O
n
e
v
e
r
y
n
a
t
u
r
a
l
t
h
i
n
g
t
o
d
o
i
s
t
o
m
i
n
i
m
i
z
e
a
c
e
r
t
a
i
n
“
e
n
e
r
g
y
f
u
n
c
t
i
o
n
”
u
n
d
e
r
t
h
e
c
o
n
s
t
r
a
i
n
t
t
h
a
t
t
h
e
i
n
t
e
r
p
o
l
a
t
i
o
n
m
u
s
t
e
v
a
l
u
a
t
e
t
o
t
h
e
g
i
v
e
n
v
a
l
u
e
s
a
t
t
h
e
r
e
f
e
r
e
n
c
e
p
o
i
n
t
s
.
G
i
v
e
n
t
h
a
t
t
h
e
e
n
e
r
g
y
f
u
n
c
t
i
o
n
i
s
c
h
o
s
e
n
w
i
s
e
l
y
,
t
h
e
r
e
s
u
l
t
i
n
g
i
n
t
e
r
p
o
l
a
t
i
o
n
w
i
l
l
n
o
t
c
o
n
t
a
i
n
a
n
y
c
o
o
r
d
i
n
a
t
e
b
i
a
s
.
O
n
e
c
a
n
t
h
i
n
k
o
f
t
h
i
s
i
n
t
e
r
p
o
l
a
t
i
o
n
a
s
a
n
e
l
a
s
t
i
c
s
p
h
e
r
e
t
h
a
t
t
r
i
e
s
t
o
m
i
n
i
m
i
z
e
i
t
s
p
o
t
e
n
t
i
a
l
e
n
e
r
g
y
.
R
e
g
u
l
a
r
i
z
a
t
i
o
n
i
s
n
o
t
n
e
e
d
e
d
i
n
t
h
i
s
i
n
t
e
r
p
o
l
a
t
i
o
n
.
P
r
o
v
i
d
e
d
t
h
a
t
t
h
e
e
n
e
r
g
y
f
u
n
c
t
i
o
n
i
s
c
h
o
s
e
n
w
e
l
l
,
h
i
g
h
-
f
r
e
q
u
e
n
c
y
m
o
d
e
s
a
r
e
p
e
n
a
l
i
z
e
d
s
u
f
f
i
c
i
e
n
t
l
y
t
o
p
r
e
v
e
n
t
e
x
t
r
e
m
e
“
s
p
i
k
i
n
e
s
s
”
.
M
o
r
e
o
v
e
r
,
w
h
i
l
e
r
e
g
u
l
a
r
i
z
a
t
i
o
n
p
r
e
v
e
n
t
s
a
n
i
n
t
e
r
p
o
l
a
t
i
o
n
f
r
o
m
e
v
a
l
u
a
t
i
n
g
e
x
a
c
t
l
y
t
o
t
h
e
g
i
v
e
n
v
a
l
u
e
s
a
t
t
h
e
r
e
f
e
r
e
n
c
e
p
o
i
n
t
s
,
w
h
i
c
h
i
n
v
a
r
i
a
b
l
y
i
n
t
r
o
d
u
c
e
s
i
n
a
c
c
u
r
a
c
i
e
s
,
t
h
e
m
i
n
i
m
i
z
a
t
i
o
n
o
f
a
n
e
n
e
r
g
y
f
u
n
c
t
i
o
n
h
a
p
p
e
n
s
u
n
d
e
r
t
h
e
c
o
n
s
t
r
a
i
n
t
t
h
a
t
t
h
e
f
u
n
c
t
i
o
n
h
a
s
e
x
a
c
t
l
y
t
h
e
r
i
g
h
t
v
a
l
u
e
s
a
t
t
h
e
r
e
f
e
r
e
n
c
e
p
o
i
n
t
s
.
T
h
e
l
a
c
k
o
f
n
e
e
d
f
o
r
r
e
g
u
l
a
r
i
z
a
t
i
o
n
a
l
s
o
s
i
m
p
l
i
f
i
e
s
t
h
e
i
n
t
e
r
p
o
l
a
t
i
o
n
p
r
o
c
e
s
s
s
i
g
n
i
f
i
c
a
n
t
l
y
,
a
s
t
h
e
u
s
e
r
d
o
e
s
n
o
t
h
a
v
e
t
o
c
h
o
o
s
e
t
h
e
r
e
g
u
l
a
r
i
z
a
t
i
o
n
p
a
r
a
m
e
t
e
r
s
.
Mathematical description
Notation and conventions
T
h
e
i
n
t
e
r
p
o
l
a
t
i
o
n
c
a
n
b
e
w
r
i
t
t
e
n
a
s
ψ
[
ϕ
,
θ
]
=
2
(
L
+
1
)
∑
j
=
1
x
j
Y
j
[
ϕ
,
θ
]
(
1
)
w
h
e
r
e
L
i
s
a
n
i
n
t
e
g
e
r
k
n
o
w
n
a
s
t
h
e
t
r
u
n
c
a
t
i
o
n
n
u
m
b
e
r
;
-
π
<
ϕ
<
=
π
a
n
d
-
π
/
2
<
=
θ
<
=
π
/
2
a
r
e
t
h
e
a
z
i
m
u
t
h
a
l
a
n
d
l
o
n
g
i
t
u
d
i
n
a
l
a
n
g
l
e
s
,
r
e
s
p
e
c
t
i
v
e
l
y
;
Y
j
a
r
e
t
h
e
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
n
u
m
b
e
r
e
d
b
y
a
s
i
n
g
l
e
n
u
m
b
e
r
s
u
c
h
t
h
a
t
Y
1
i
s
0
Y
0
(
t
h
e
c
o
n
s
t
a
n
t
f
u
n
c
t
i
o
n
)
a
n
d
t
h
e
o
t
h
e
r
s
c
a
n
g
o
i
n
a
n
y
o
r
d
e
r
;
x
j
a
r
e
t
h
e
u
n
k
n
o
w
n
c
o
e
f
f
i
c
i
e
n
t
s
.
T
h
e
f
o
r
m
u
l
a
(
1
)
d
e
s
c
r
i
b
e
s
a
l
l
p
o
s
s
i
b
l
e
l
i
n
e
a
r
c
o
m
b
i
n
a
t
i
o
n
s
o
f
t
h
e
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
k
Y
l
w
i
t
h
l
<
=
L
;
t
h
e
u
p
p
e
r
l
i
m
i
t
o
f
s
u
m
m
a
t
i
o
n
2
(
L
+
1
)
c
o
m
e
s
f
r
o
m
a
d
d
i
n
g
u
p
t
h
e
n
u
m
b
e
r
o
f
h
a
r
m
o
n
i
c
s
w
i
t
h
d
i
f
f
e
r
e
n
t
l
’
s
:
1
+
3
+
5
+
.
.
.
+
(
1
+
2
L
)
=
2
(
L
+
1
)
.
T
h
e
r
e
q
u
i
r
e
m
e
n
t
t
h
a
t
t
h
e
i
n
t
e
r
p
o
l
a
t
i
o
n
e
v
a
l
u
a
t
e
s
t
o
t
h
e
g
i
v
e
n
v
a
l
u
e
s
a
t
t
h
e
r
e
f
e
r
e
n
c
e
p
o
i
n
t
s
c
a
n
b
e
w
r
i
t
t
e
n
i
n
m
a
t
r
i
x
f
o
r
m
a
s
A
x
=
b
(
2
)
w
h
e
r
e
b
i
=
ψ
i
i
s
t
h
e
(
k
n
o
w
n
)
v
a
l
u
e
o
f
t
h
e
i
n
t
e
r
p
o
l
a
t
e
d
f
u
n
c
t
i
o
n
a
t
t
h
e
i
-
t
h
p
o
i
n
t
.
A
i
j
=
Y
j
[
ϕ
i
,
θ
i
]
i
s
t
h
e
v
a
l
u
e
o
f
t
h
e
j
-
t
h
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
a
t
t
h
e
i
-
t
h
p
o
i
n
t
.
(
I
f
A
w
e
r
e
s
q
u
a
r
e
,
w
e
w
o
u
l
d
c
a
l
l
i
t
t
h
e
d
e
s
i
g
n
m
a
t
r
i
x
)
.
B
e
c
a
u
s
e
t
h
e
b
a
s
i
s
f
u
n
c
t
i
o
n
s
a
r
e
d
i
f
f
e
r
e
n
t
,
A
i
s
f
u
l
l
-
r
a
n
k
.
L
e
t
u
s
d
e
n
o
t
e
m
t
h
e
n
u
m
b
e
r
o
f
d
a
t
a
p
o
i
n
t
s
a
n
d
n
=
2
(
L
+
1
)
t
h
e
n
u
m
b
e
r
o
f
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
u
s
e
d
f
o
r
i
n
t
e
r
p
o
l
a
t
i
o
n
.
T
h
e
n
,
t
h
e
m
a
t
r
i
x
A
i
s
m
n
.
T
o
e
n
s
u
r
e
t
h
a
t
t
h
e
m
o
d
e
l
i
s
n
o
t
u
n
d
e
r
d
e
t
e
r
m
i
n
e
d
,
n
>
=
m
.
T
h
e
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
i
n
t
h
i
s
e
s
s
a
y
a
r
e
d
e
f
i
n
e
d
a
s
f
o
l
l
o
w
s
:
k
Y
l
≡
k
N
l
C
o
s
[
k
ϕ
]
k
P
l
[
S
i
n
[
θ
]
]
k
>
=
0
S
i
n
[
|
k
|
ϕ
]
|
k
|
P
l
[
S
i
n
[
θ
]
]
k
<
0
(
3
)
w
h
e
r
e
k
N
l
i
s
a
n
o
r
m
a
l
i
z
a
t
i
o
n
f
a
c
t
o
r
a
n
d
k
P
l
i
s
t
h
e
a
s
s
o
c
i
a
t
e
d
L
e
g
e
n
d
r
e
p
o
l
y
n
o
m
i
a
l
.
T
h
e
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
s
a
s
d
e
f
i
n
e
d
b
y
(
3
)
a
r
e
e
i
g
e
n
f
u
n
c
t
i
o
n
s
o
f
t
h
e
L
a
p
l
a
c
e
o
p
e
r
a
t
o
r
w
i
t
h
e
i
g
e
n
v
a
l
u
e
-
l
(
l
+
1
)
a
n
d
o
f
t
h
e
o
p
e
r
a
t
o
r
2
∂
ϕ
w
i
t
h
e
i
g
e
n
v
a
l
u
e
-
2
k
;
t
h
e
y
a
r
e
a
l
s
o
o
r
t
h
o
n
o
r
m
a
l
.
T
h
e
r
e
a
s
o
n
t
o
r
e
p
l
a
c
e
ϕ
w
i
t
h
s
i
n
e
s
a
n
d
c
o
s
i
n
e
s
i
s
t
o
m
a
k
e
t
h
e
b
a
s
i
s
f
u
n
c
t
i
o
n
s
r
e
a
l
(
s
i
n
c
e
t
h
i
s
e
s
s
a
y
w
i
l
l
o
n
l
y
c
o
n
s
i
d
e
r
i
n
t
e
r
p
o
l
a
t
i
n
g
r
e
a
l
f
u
n
c
t
i
o
n
s
)
.
T
h
e
n
u
m
b
e
r
i
n
g
u
s
e
d
i
n
t
h
e
e
s
s
a
y
i
s
Y
1
=
0
Y
0
,
Y
2
=
-
1
Y
1
,
Y
3
=
0
Y
1
,
Y
4
=
+
1
Y
1
,
Y
5
=
-
2
Y
2
,
Y
6
=
-
1
Y
2
,
a
n
d
s
o
o
n
;
b
u
t
a
g
a
i
n
,
t
h
e
o
r
d
e
r
d
o
e
s
n
’
t
r
e
a
l
l
y
c
h
a
n
g
e
a
n
y
t
h
i
n
g
,
t
h
e
o
n
l
y
c
o
n
v
e
n
t
i
o
n
t
h
a
t
m
u
s
t
b
e
k
e
p
t
i
s
t
h
a
t
Y
1
=
0
Y
0
.
Resolving the overdetermined system
W
h
e
n
a
m
o
d
e
l
h
a
s
m
o
r
e
p
a
r
a
m
e
t
e
r
s
t
h
a
n
c
o
n
s
t
r
a
i
n
t
s
,
o
n
e
p
o
s
s
i
b
l
e
s
o
l
u
t
i
o
n
i
s
t
o
u
s
e
t
h
e
e
x
t
r
a
d
e
g
r
e
e
s
o
f
f
r
e
e
d
o
m
t
o
m
i
n
i
m
i
z
e
a
c
e
r
t
a
i
n
e
n
e
r
g
y
f
u
n
c
t
i
o
n
.
O
n
e
w
a
y
t
o
d
o
t
h
a
t
i
s
t
o
e
x
p
r
e
s
s
t
h
e
n
-
c
o
m
p
o
n
e
n
t
v
e
c
t
o
r
x
i
n
t
e
r
m
s
o
f
a
s
m
a
l
l
e
r
m
-
c
o
m
p
o
n
e
n
t
v
e
c
t
o
r
u
s
u
c
h
t
h
a
t
a
n
y
v
e
c
t
o
r
x
c
o
m
p
u
t
e
d
f
r
o
m
u
w
o
u
l
d
a
u
t
o
m
a
t
i
c
a
l
l
y
m
i
n
i
m
i
z
e
t
h
e
e
n
e
r
g
y
.
T
h
i
s
w
a
y
,
w
e
w
o
u
l
d
h
a
v
e
m
u
n
k
n
o
w
n
c
o
m
p
o
n
e
n
t
s
o
f
u
t
o
s
o
l
v
e
e
x
a
c
t
l
y
m
e
q
u
a
t
i
o
n
s
.
T
o
m
a
k
e
t
h
e
m
i
n
i
m
i
z
a
t
i
o
n
e
q
u
a
t
i
o
n
s
l
i
n
e
a
r
,
l
e
t
u
s
o
n
l
y
c
o
n
s
i
d
e
r
q
u
a
d
r
a
t
i
c
e
n
e
r
g
y
f
u
n
c
t
i
o
n
s
.
I
f
ψ
=
x
j
Y
j
,
t
h
e
n
e
n
e
r
g
y
i
s
g
i
v
e
n
b
y
E
=
T
x
W
0
x
(
4
)
w
h
e
r
e
W
0
i
s
a
s
y
m
m
e
t
r
i
c
m
a
t
r
i
x
o
f
w
e
i
g
h
t
s
.
T
h
u
s
,
t
h
e
p
r
o
b
l
e
m
i
s
m
i
n
i
m
i
z
i
n
g
t
h
e
e
n
e
r
g
y
(
4
)
w
h
i
l
e
o
b
e
y
i
n
g
t
h
e
e
q
u
a
t
i
o
n
A
x
=
b
.
I
n
o
t
h
e
r
w
o
r
d
s
,
i
f
t
h
e
c
o
e
f
f
i
c
i
e
n
t
v
e
c
t
o
r
i
s
x
,
t
h
e
n
t
h
e
v
a
l
u
e
o
f
e
n
e
r
g
y
m
u
s
t
n
o
t
c
h
a
n
g
e
(
t
o
f
i
r
s
t
o
r
d
e
r
)
i
f
t
h
e
c
o
e
f
f
i
c
i
e
n
t
v
e
c
t
o
r
w
e
r
e
c
h
a
n
g
e
d
t
o
x
-
>
x
+
ε
δ
,
w
i
t
h
ε
-
>
0
,
f
o
r
a
n
y
v
e
c
t
o
r
δ
a
l
l
o
w
e
d
b
y
t
h
e
c
o
n
s
t
r
a
i
n
t
A
x
=
b
.
M
a
t
h
e
m
a
t
i
c
a
l
l
y
,
t
h
e
g
r
a
d
i
e
n
t
o
f
E
i
s
o
r
t
h
o
g
o
n
a
l
t
o
e
v
e
r
y
v
e
c
t
o
r
i
n
t
h
e
n
u
l
l
s
p
a
c
e
o
f
A
.
S
i
n
c
e
t
h
e
o
r
t
h
o
g
o
n
a
l
c
o
m
p
l
e
m
e
n
t
o
f
t
h
e
n
u
l
l
s
p
a
c
e
i
s
t
h
e
r
o
w
s
p
a
c
e
,
t
h
e
g
r
a
d
i
e
n
t
o
f
E
m
u
s
t
l
i
e
i
n
t
h
e
r
o
w
s
p
a
c
e
o
f
A
.
T
h
e
s
a
m
e
c
o
n
c
l
u
s
i
o
n
c
a
n
b
e
r
e
a
c
h
e
d
u
s
i
n
g
L
a
g
r
a
n
g
e
’
s
m
u
l
t
i
p
l
i
e
r
s
.
W
0
x
∈
R
[
A
]
(
5
)
G
i
v
e
n
t
h
a
t
m
o
s
t
p
h
y
s
i
c
a
l
a
n
d
r
e
a
l
-
w
o
r
l
d
f
u
n
c
t
i
o
n
s
a
r
e
d
e
f
i
n
e
d
u
p
t
o
a
c
o
n
s
t
a
n
t
-
f
o
r
e
x
a
m
p
l
e
,
t
e
m
p
e
r
a
t
u
r
e
-
t
h
e
e
n
e
r
g
y
f
u
n
c
t
i
o
n
m
u
s
t
n
o
t
p
u
n
i
s
h
c
o
n
s
t
a
n
t
f
u
n
c
t
i
o
n
s
.
I
n
o
t
h
e
r
w
o
r
d
s
,
a
d
d
i
n
g
0
Y
0
(
a
c
o
n
s
t
a
n
t
)
t
o
ψ
s
h
o
u
l
d
n
o
t
c
h
a
n
g
e
e
n
e
r
g
y
.
W
h
a
t
t
h
i
s
m
e
a
n
s
i
s
t
h
a
t
t
h
e
f
i
r
s
t
c
o
l
u
m
n
a
n
d
r
o
w
o
f
W
0
s
h
o
u
l
d
b
e
z
e
r
o
.
W
0
t
h
e
n
h
a
s
t
h
e
(
s
o
m
e
w
h
a
t
p
r
o
b
l
e
m
a
t
i
c
)
f
o
r
m
W
0
=
0
0
0
W
(
6
)
T
h
e
e
q
u
a
t
i
o
n
a
b
o
v
e
i
s
w
r
i
t
t
e
n
i
n
b
l
o
c
k
m
a
t
r
i
x
n
o
t
a
t
i
o
n
;
h
e
r
e
,
t
h
e
t
o
p
-
l
e
f
t
z
e
r
o
i
s
j
u
s
t
t
h
e
n
u
m
b
e
r
0
,
t
h
e
o
f
f
-
d
i
a
g
o
n
a
l
z
e
r
o
s
d
e
n
o
t
e
b
l
o
c
k
s
o
f
n
-
1
z
e
r
o
s
,
a
n
d
t
h
e
m
a
t
r
i
x
W
i
s
t
h
e
“
i
n
n
e
r
p
a
r
t
”
o
f
W
0
.
O
t
h
e
r
t
h
a
n
t
h
e
c
o
n
s
t
a
n
t
f
u
n
c
t
i
o
n
,
t
h
e
r
e
i
s
n
o
f
u
n
c
t
i
o
n
t
h
a
t
c
a
n
b
e
a
d
d
e
d
t
o
ψ
s
u
c
h
t
h
a
t
E
d
o
e
s
n
o
t
c
h
a
n
g
e
.
C
o
r
r
e
s
p
o
n
d
i
n
g
l
y
,
t
h
e
n
u
l
l
s
p
a
c
e
o
f
W
i
s
j
u
s
t
t
h
e
z
e
r
o
v
e
c
t
o
r
,
w
h
i
c
h
m
e
a
n
s
t
h
e
m
a
t
r
i
x
i
s
i
n
v
e
r
t
i
b
l
e
;
t
h
i
s
i
s
g
o
i
n
g
t
o
b
e
i
m
p
o
r
t
a
n
t
.
T
h
e
e
q
u
a
t
i
o
n
(
5
)
b
e
c
o
m
e
s
0
0
0
W
x
∈
R
[
A
]
(
7
)
W
e
s
e
e
t
h
a
t
t
h
e
f
i
r
s
t
e
n
t
r
y
x
1
o
f
t
h
e
v
e
c
t
o
r
x
d
o
e
s
n
o
t
p
l
a
y
a
n
y
r
o
l
e
i
n
t
h
e
e
q
u
a
t
i
o
n
(
7
)
.
A
n
a
t
u
r
a
l
t
h
i
n
g
t
o
d
o
i
s
t
o
s
e
p
a
r
a
t
e
t
h
e
v
e
c
t
o
r
i
n
t
o
t
w
o
c
o
m
p
o
n
e
n
t
s
:
x
=
x
1
y
(
8
)
w
h
e
r
e
b
l
o
c
k
m
a
t
r
i
x
n
o
t
a
t
i
o
n
i
s
u
s
e
d
:
x
1
i
s
a
n
u
m
b
e
r
a
n
d
y
i
s
a
n
n
-
1
c
o
m
p
o
n
e
n
t
v
e
c
t
o
r
.
T
h
e
e
q
u
a
t
i
o
n
(
7
)
t
h
e
n
b
e
c
o
m
e
s
0
W
y
∈
R
[
A
]
,
o
r
,
e
q
u
i
v
a
l
e
n
t
l
y
,
0
W
y
=
T
A
v
(
9
)
f
o
r
s
o
m
e
m
-
c
o
m
p
o
n
e
n
t
v
e
c
t
o
r
v
.
T
h
e
m
a
t
r
i
x
A
w
a
s
d
e
f
i
n
e
d
a
s
A
i
j
=
Y
j
[
ϕ
i
,
θ
i
]
-
i
n
o
t
h
e
r
w
o
r
d
s
,
t
h
e
j
-
t
h
c
o
l
u
m
n
o
f
A
l
i
s
t
s
t
h
e
v
a
l
u
e
s
o
f
t
h
e
j
-
t
h
s
p
h
e
r
i
c
a
l
h
a
r
m
o
n
i
c
a
t
t
h
e
r
e
f
e
r
e
n
c
e
p
o
i
n
t
s
.
A
i
1
,
i
n
p
a
r
t
i
c
u
l
a
r
,
l
i
s
t
s
t
h
e
v
a
l
u
e
s
o
f
Y
1
;
b
u
t
Y
1
=
0
Y
0
i
s
a
c
o
n
s
t
a
n
t
,
s
o
A
i
1
i
s
t
h
e
s
a
m
e
f
o
r
a
l
l
i
.
A
c
o
n
v
e
n
i
e
n
t
w
a
y
t
o
w
r
i
t
e
t
h
i
s
i
s
a
s
f
o
l
l
o
w
s
:
A
=
(
c
B
)
(
1
0
)
w
h
e
r
e
c
i
s
a
n
m
1
c
o
l
u
m
n
v
e
c
t
o
r
o
f
t
h
e
s
a
m
e
n
u
m
b
e
r
Y
1
a
n
d
B
i
s
a
n
m
(
n
-
1
)
m
a
t
r
i
x
.
T
h
e
e
q
u
a
t
i
o
n
(
9
)
t
h
e
n
b
e
c
o
m
e
s
0
W
y
=
T
c
T
B
v
(
1
1
)
B
y
l
o
o
k
i
n
g
a
t
t
h
e
f
i
r
s
t
r
o
w
o
f
t
h
e
e
q
u
a
t
i
o
n
,
w
e
g
e
t
T
c
v
=
0
.
O
n
e
w
a
y
t
o
a
c
h
i
e
v
e
t
h
i
s
e
q
u
a
l
i
t
y
i
s
t
o
d
e
c
l
a
r
e
v
i
=
u
i
f
o
r
i
>
=
2
a
n
d
v
1
=
-
m
∑
i
=
2
u
i
f
o
r
s
o
m
e
o
t
h
e
r
v
e
c
t
o
r
u
.
I
n
t
e
r
m
s
o
f
t
h
e
v
e
c
t
o
r
u
,
t
h
e
s
e
c
o
n
d
p
a
r
t
o
f
(
1
1
)
b
e
c
o
m
e
s
W
y
=
T
B
0
-
1
-
1
-
1
.
.
.
1
1
1
.
.
.
u
(
1
2
)
w
h
e
r
e
t
h
e
u
n
n
a
m
e
d
m
a
t
r
i
x
i
s
o
f
s
i
z
e
m
m
a
n
d
a
l
l
t
h
e
u
n
f
i
l
l
e
d
s
l
o
t
s
a
r
e
z
e
r
o
;
i
t
c
a
n
b
e
w
r
i
t
t
e
n
i
n
p
i
e
c
e
w
i
s
e
n
o
t
a
t
i
o
n
a
s
[
u
n
n
a
m
e
d
m
a
t
r
i
x
]
i
j
=
δ
i
j
-
δ
i
1
.
T
h
e
f
i
r
s
t
c
o
m
p
o
n
e
n
t
o
f
u
i
s
n
o
t
u
s
e
d
h
e
r
e
,
s
o
w
e
c
a
n
u
s
e
i
t
t
o
s
e
t
x
1
=
u
1
i
n
t
h
e
f
o
r
m
u
l
a
(
8
)
.
T
h
u
s
,
w
e
h
a
v
e
x
=
M
u
(
1
3
)
w
h
e
r
e
M
=
1
0
0
F
(
1
4
)
I
n
t
h
e
f
o
r
m
u
l
a
(
1
4
)
,
t
h
e
1
i
n
t
h
e
t
o
p
l
e
f
t
c
o
r
n
e
r
i
s
t
h
e
n
u
m
b
e
r
1
,
F
i
s
a
n
(
n
-
1
)
(
m
-
1
)
m
a
t
r
i
x
,
a
n
d
t
h
e
z
e
r
o
s
r
e
p
r
e
s
e
n
t
b
l
o
c
k
s
o
f
z
e
r
o
s
o
f
a
p
p
r
o
p
r
i
a
t
e
s
i
z
e
.
T
h
e
f
o
r
m
u
l
a
f
o
r
F
i
s
a
s
f
o
l
l
o
w
s
:
F
=
-
1
W
T
B
-
1
-
1
-
1
.
.
.
1
1
1
.
.
.
(
1
5
)
B
y
c
o
m
p
o
n
e
n
t
s
,
F
i
j
=
(
-
1
W
T
B
)
i
,
j
+
1
-
(
-
1
W
T
B
)
i
,
1
.
T
h
e
u
n
n
a
m
e
d
m
a
t
r
i
x
i
s
o
f
s
i
z
e
m
(
m
-
1
)
;
i
t
i
s
l
i
k
e
t
h
e
u
n
n
a
m
e
d
m
a
t
r
i
x
i
n
(
1
2
)
b
u
t
w
i
t
h
t
h
e
f
i
r
s
t
c
o
l
u
m
n
r
e
m
o
v
e
d
.
I
t
c
a
n
b
e
s
h
o
w
n
t
h
a
t
w
h
a
t
e
v
e
r
u
i
s
p
a
s
s
e
d
i
n
t
o
t
h
e
f
o
r
m
u
l
a
(
1
3
)
,
t
h
e
r
e
s
u
l
t
i
s
s
u
c
h
t
h
a
t
W
0
x
∈
R
[
A
]
.
W
e
h
a
v
e
e
f
f
e
c
t
i
v
e
l
y
t
a
k
e
n
c
a
r
e
o
f
t
h
e
m
i
n
i
m
i
z
a
t
i
o
n
p
a
r
t
o
f
t
h
e
p
r
o
b
l
e
m
a
n
d
n
o
w
o
n
l
y
n
e
e
d
t
o
s
a
t
i
s
f
y
A
x
=
b
b
y
f
i
n
d
i
n
g
t
h
e
r
i
g
h
t
u
.
T
h
e
v
e
c
t
o
r
u
h
a
s
m
c
o
m
p
o
n
e
n
t
s
-
t
h
i
s
i
s
j
u
s
t
e
n
o
u
g
h
t
o
s
a
t
i
s
f
y
t
h
e
m
c
o
n
s
t
r
a
i
n
t
s
.
S
u
b
s
t
i
t
u
t
i
n
g
x
=
M
u
i
n
t
o
A
x
=
b
,
w
e
h
a
v
e
A
M
u
=
b
(
1
6
)
A
M
c
a
n
b
e
w
r
i
t
t
e
n
a
s
(
c
B
F
)
,
w
h
e
r
e
c
d
e
n
o
t
e
s
a
n
n
1
v
e
c
t
o
r
o
f
Y
1
.
B
e
c
a
u
s
e
W
i
s
s
y
m
m
e
t
r
i
c
,
-
1
W
T
B
=
T
(
B
-
1
W
)
;
d
e
n
o
t
i
n
g
C
≡
B
-
1
W
,
w
e
c
a
n
e
f
f
i
c
i
e
n
t
l
y
c
o
m
p
u
t
e
F
a
s
F
i
j
=
C
j
+
1
,
i
-
C
1
,
i
.
F
i
n
a
l
l
y
,
t
h
e
v
e
c
t
o
r
x
c
a
n
b
e
c
o
m
p
u
t
e
d
b
y
s
u
b
s
t
i
t
u
t
i
n
g
u
i
n
t
o
(
1
3
)
.
I
t
s
o
l
v
e
s
A
x
=
b
a
n
d
u
s
e
s
t
h
e
r
e
m
a
i
n
i
n
g
d
e
g
r
e
e
s
o
f
f
r
e
e
d
o
m
t
o
m
i
n
i
m
i
z
e
T
x
W
0
x
.
N
u
m
e
r
i
c
a
l
c
h
e
c
k
Relation to Tikhonov regularization
T
h
e
f
o
r
m
u
l
a
(
1
6
)
b
a
s
i
c
a
l
l
y
m
i
n
i
m
i
z
e
s
t
h
e
q
u
a
n
t
i
t
y
T
(
A
x
-
b
)
(
A
x
-
b
)
+
λ
T
x
W
0
x
,
w
h
e
r
e
λ
-
>
+
0
.
T
h
i
s
c
a
n
b
e
s
e
e
n
f
r
o
m
a
p
h
y
s
i
c
a
l
m
e
t
a
p
h
o
r
o
f
a
s
m
a
l
l
b
e
a
d
i
n
s
i
d
e