aarjaneiro/planetmath huggingface dataset:
cubic spline interpolation

cubic spline interpolation

Suppose we are given

N

+

1

data points

{

(

x

k

,

y

k

)

}

such
that

a

=

x

0

<

…

<

x

N

.

(1)

Then the function

S

⁢

(

x

)

is called a

cubic spline interpolation

if
there exists

N

cubic polynomials

S

k

⁢

(

x

)

with coefficients

s

k

,

i

⁢

0

≤

i

≤

3

such that the following hold.

1.

S

⁢

(

x

)

=

S

k

⁢

(

x

)

=

∑

i

=

0

3

s

k

,

i

⁢

(

x

-

x

k

)

i

⁢

∀

x

∈

[

x

k

,

x

k

+

1

]

⁢

0

≤

k

≤

N

-

1

2.

S

⁢

(

x

k

)

=

y

k

⁢

0

≤

k

≤

N

3.

S

k

⁢

(

x

k

+

1

)

=

S

k

+

1

⁢

(

x

k

+

1

)

⁢

0

≤

k

≤

N

-

2

4.

S

k

′

⁢

(

x

k

+

1

)

=

S

k

+

1

′

⁢

(

x

k

+

1

)

⁢

0

≤

k

≤

N

-

2

5.

S

k

′′

⁢

(

x

k

+

1

)

=

S

k

+

1

′′

⁢

(

x

k

+

1

)

⁢

0

≤

k

≤

N

-

2

The set of points

(

⁢

1

⁢

)

are called the knots. The set of
cubic splines on a fixed set of knots, forms a

vector space

for
cubic spline

addition

and

scalar multiplication

.

So we see that the cubic spline not only interpolates the data

{

(

x

k

,

y

k

)

}

but matches the first and second derivatives
at the knots. Notice, from the above

definition

, one is free to
specify constraints on the

endpoints

. One common end point
constraint is

S

′′

⁢

(

a

)

=

0

⁢

S

′′

⁢

(

b

)

=

0

,
which is called the natural spline. Other popular choices are the
clamped cubic spline, parabolically terminated spline and
curvature-adjusted spline. Cubic splines are frequently used in
numerical analysis to fit data. Matlab uses the command spline to
find cubic spline interpolations with not-a-knot end point
conditions. For example, the following commands would find the
cubic spline interpolation of the curve

4

⁢

cos

⁡

(

x

)

+

1

and plot the
curve and the

interpolation

marked with

o’s

.

x = 0:2*pi;
y = 4*cos(x)+1;
xx = 0:.001:2*pi;
yy = spline(x,y,xx);
plot(x,y,'o',xx,yy)

Title

cubic spline interpolation

Canonical

name

CubicSplineInterpolation

Date of creation

2013-03-22 13:40:25

Last modified on

2013-03-22 13:40:25

Owner

yota (10184)

Last modified by

yota (10184)

Numerical id

7

Author

yota (10184)

Entry type

Definition

Classification

msc 65-01

Generated on Sat Feb 10 12:34:52 2018 by

LaTeXML
Euler-Maclaurin summation formula

Euler-Maclaurin summation formula

Let

B

r

be the

r

⁢

th

Bernoulli number

, and

B

r

⁢

(

x

)

be the

r

⁢

th

Bernoulli periodic function

. For any

integer

k

≥

0

and for any function

f

of class

C

k

+

1

on

[

a

,

b

]

,

a

,

b

∈

ℤ

, we have

∑

a

<

n

≤

b

f

⁢

(

n

)

=

∫

a

b

f

⁢

(

t

)

⁢

𝑑

t

+

∑

r

=

0

k

(

-

1

)

r

+

1

⁢

B

r

+

1

(

r

+

1

)

!

⁢

(

f

(

r

)

⁢

(

b

)

-

f

(

r

)

⁢

(

a

)

)

+

(

-

1

)

k

(

k

+

1

)

!

⁢

∫

a

b

B

k

+

1

⁢

(

t

)

⁢

f

(

k

+

1

)

⁢

(

t

)

⁢

𝑑

t

.

Title

Euler-Maclaurin summation formula

Canonical

name

EulerMaclaurinSummationFormula

Date of creation

2013-03-22 11:46:01

Last modified on

2013-03-22 11:46:01

Owner

KimJ (5)

Last modified by

KimJ (5)

Numerical id

9

Author

KimJ (5)

Entry type

Theorem

Classification

msc 65B15

Classification

msc 00-02

Related topic

BernoulliNumber

Generated on Sat Feb 10 12:34:54 2018 by

LaTeXML
computing powers by repeated squaring

computing powers by repeated squaring

From time to time, there arise occasions where one wants to raise a
number to a large integer power. While one can compute

x

n

by
multiplying

x

by itself

n

times, not only does this entail a
large expenditure of effort when

n

is large, but roundoff errors
can accumulate. A more efficient approach is to first rise

x

to
powers of two by successive squaring, then multiply to obtain

x

n

.
This procedure will now be explained by computing

1.08145

187

.

The first step is to express

n

as a sum of powers of two, i.e. in
base 2. In our example, we have

187

=

1

+

2

+

8

+

16

+

32

+

128

.

By the basic properties of exponentiation, we therefore have

x

187

=

x

⋅

x

2

⋅

x

8

⋅

x

16

⋅

x

32

⋅

x

128

.

The next step is to raise 1.08145 to powers of two, which we accomplish
by repeatedly squaring like so: (In order to guard against roundoff
error, we work with more decimal places than needed and round off at the
end of the

computation

.)

x

=

1.08145

x

2

=

1.1695341

x

4

=

(

x

2

)

2

=

1.1695341

2

=

1.3678100

x

8

=

(

x

4

)

2

=

1.3678100

2

=

1.8709042

x

16

=

(

x

8

)

2

=

1.8709042

2

=

3.5002825

x

32

=

(

x

16

)

2

=

3.5002825

2

=

12.251978

x

64

=

(

x

32

)

2

=

12.251978

2

=

150.110965

x

128

=

(

x

64

)

2

=

150.110965

2

=

26523642.95

Finally, we multiply together the appropriate powers to obtain our
answer:

1.08145

⋅

1.1695341

⋅

1.8709042

⋅

3.5002825

⋅

12.251978

⋅

26523642.95

=

2691617615

Hence, we compute

1.08145

187

=

2.69162

×

10

9

.

Note that this computation involved only 12 multiplications
as opposed to 186 multiplications. More generally, we will
require

log

2

⁡

n

repeated squarings and up to

log

2

⁡

n

multiplications of the repeated squares, or a total of between

log

2

⁡

n

and

2

⁢

log

2

⁡

n

multiplications to compute

x

n

this way.

More generally, the same method can be used to compute

x

n

when

x

is a complex number or a matrix, or something even
more general, such as an

element of

a group. Thus this
approach can be adapted to computing

exponentials

and

trigonometric functions

, numerical approximation of

differential
equation

, and the like.

Title

computing powers by repeated squaring

Canonical name

ComputingPowersByRepeatedSquaring

Date of creation

2013-03-22 19:18:45

Last modified on

2013-03-22 19:18:45

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Algorithm

Classification

msc 65B99

Generated on Sat Feb 10 12:35:01 2018 by

LaTeXML
truly random numbers

truly random numbers

Truly

random numbers

can only be generated by a physical process and cannot be generated via software. This makes it rather clumsy to use them in

Monte Carlo

calculations, since they must be first generated in a separate device and either sent to the

computer

or recorded (for example on removable storage media) for later use in calculations. Traditionally, tapes containing millions of random numbers generated using radioactive decay were available from laboratories.

Nowadays, standard digital computers often have provisions for obtaining truly random numbers, that is, numbers generated by a physical process. Fir instance, Intel has provided a since their

i

⁢

810

chipsets which utilizes noise in a particularly prone semiconductor as a source of randomness. Often times it is possible to use other incidental physical noise as a source; for example, static on the input channel of a sound card. In

addition

, peripheral devices (add-ons) to personal computers exist which provide truly random numbers, when the previous methods fail.

References

1

Derived from) The Data Analysis Briefbook.

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html.

Title

truly random numbers

Canonical

name

TrulyRandomNumbers

Date of creation

2013-03-22 12:07:16

Last modified on

2013-03-22 12:07:16

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

7

Author

akrowne (2)

Entry type

Definition

Classification

msc 65C10

Classification

msc 11K45

Related topic

PseudorandomNumbers

Generated on Sat Feb 10 12:35:11 2018 by

LaTeXML
Lagrange interpolation formula

Lagrange interpolation formula

Let

(

x

1

,

y

1

)

,

(

x

2

,

y

2

)

,

…

,

(

x

n

,

y

n

)

be

n

points in the plane (

x

i

≠

x

j

for

i

≠

j

). Then there exists a unique

polynomial

p

⁢

(

x

)

of degree at most

n

-

1

such that

y

i

=

p

⁢

(

x

i

)

for

i

=

1

,

…

,

n

.

Such polynomial can be found using

Lagrange’s interpolation formula

:

p

⁢

(

x

)

=

f

⁢

(

x

)

(

x

-

x

1

)

⁢

f

′

⁢

(

x

1

)

⁢

y

1

+

f

⁢

(

x

)

(

x

-

x

2

)

⁢

f

′

⁢

(

x

2

)

⁢

y

2

+

⋯

+

f

⁢

(

x

)

(

x

-

x

n

)

⁢

f

′

⁢

(

x

n

)

⁢

y

n

where

f

⁢

(

x

)

=

(

x

-

x

1

)

⁢

(

x

-

x

2

)

⁢

⋯

⁢

(

x

-

x

n

)

.

To see this, notice that the above formula is the same as

p

⁢

(

x

)

=

y

1

⁢

(

x

-

x

2

)

⁢

(

x

-

x

3

)

⁢

…

⁢

(

x

-

x

n

)

(

x

1

-

x

2

)

⁢

(

x

1

-

x

3

)

⁢

…

⁢

(

x

1

-

x

n

)

+

y

2

⁢

(

x

-

x

1

)

⁢

(

x

-

x

3

)

⁢

…

⁢

(

x

-

x

n

)

(

x

2

-

x

1

)

⁢

(

x

2

-

x

3

)

⁢

…

⁢

(

x

2

-

x

n

)

+

…

+

y

n

⁢

(

x

-

x

1

)

⁢

(

x

-

x

2

)

⁢

…

⁢

(

x

-

x

n

-

1

)

(

x

n

-

x

1

)

⁢

(

x

n

-

x

2

)

⁢

…

⁢

(

x

n

-

x

n

-

1

)

and that for all

x

i

, every

numerator

except one vanishes, and this numerator will be identical to the denominator, making the overall

quotient

equal to 1. Therefore, each

p

⁢

(

x

i

)

equals

y

i

.

Title

Lagrange interpolation formula

Canonical name

LagrangeInterpolationFormula

Date of creation

2013-03-22 11:46:21

Last modified on

2013-03-22 11:46:21

Owner

drini (3)

Last modified by

drini (3)

Numerical id

16

Author

drini (3)

Entry type

Theorem

Classification

msc 65D05

Classification

msc 41A05

Synonym

Lagrange’s Interpolation formula

Related topic

SimpsonsRule

Related topic

LectureNotesOnPolynomialInterpolation

Defines

Lagrange polynomial

Generated on Sat Feb 10 12:35:13 2018 by

LaTeXML
proof of uniqueness of Lagrange Interpolation formula

proof of uniqueness of Lagrange Interpolation formula

Existence

is clear from the construction, the uniqueness is proved by assuming there are two different

polynomials

p

⁢

(

x

)

and

q

⁢

(

x

)

that interpolate the points. Then

r

⁢

(

x

)

=

p

⁢

(

x

)

-

q

⁢

(

x

)

has

n

zeros,

x

1

,

…

,

x

n

and there is a point

x

e

such that

r

⁢

(

x

e

)

≠

0

.

r

⁢

(

x

)

is non-constant with degree

deg

⁡

(

r

⁢

(

x

)

)

≤

n

-

1

and has more than

n

-

1

solutions, which is a

contradiction

. Thus there can only be one polynomial.

Title

proof of uniqueness of Lagrange Interpolation formula

Canonical name

ProofOfUniquenessOfLagrangeInterpolationFormula

Date of creation

2013-03-22 14:09:25

Last modified on

2013-03-22 14:09:25

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 65D05

Classification

msc 41A05

Generated on Sat Feb 10 12:35:23 2018 by

LaTeXML
Vandermonde interpolation approach

Vandermonde interpolation approach

The Vandermonde approach for

interpolation

is when we wish to determine the interpolating polynomial

p

⁢

(

x

)

=

a

0

+

a

1

⁢

x

+

a

2

⁢

x

2

+

…

+

a

n

⁢

x

n

for the

n

+

1

points

(

x

i

,

y

i

)

,

i

=

0

,

1

,

…

,

n

by forming the equations

y

i

=

a

0

+

a

1

⁢

x

i

+

a

2

⁢

x

2

2

+

…

+

a

n

⁢

x

n

n

for

i

=

0

,

1

,

…

,

n

, and solving for the unknown coefficients

a

0

,

a

1

,

…

,

a

n

.

The system of equations can be written by using matrices

Y

=

X

⁢

A

where

X

is a

Vandermonde matrix

.

Title

Vandermonde interpolation approach

Canonical name

VandermondeInterpolationApproach

Date of creation

2013-03-22 14:19:56

Last modified on

2013-03-22 14:19:56

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 65D05

Classification

msc 41A05

Generated on Sat Feb 10 12:35:25 2018 by

LaTeXML
Bernstein polynomial

Bernstein polynomial

The

Bernstein polynomials

of

n

are defined by

B

i

n

⁢

(

t

)

=

(

n

i

)

⁢

t

i

⁢

(

1

-

t

)

n

-

i

i

=

0

,

1

,

2

,

…

,

n

where

(

n

i

)

is the

binomial coefficient

.

Bernstein polynomials are used extensively in

interpolation

theory and in computer graphics. They can be computed efficiently using the de Casteljau’s algorithm.

References

1

Gerald Farin,

Curves and Surfaces for CAGD, A Practical Guide, 5th edition

,
Academic Press, 2002.

Title

Bernstein polynomial

Canonical name

BernsteinPolynomial

Date of creation

2013-03-22 14:15:25

Last modified on

2013-03-22 14:15:25

Owner

stitch (17269)

Last modified by

stitch (17269)

Numerical id

9

Author

stitch (17269)

Entry type

Definition

Classification

msc 65D17

Synonym

Bernstein basis polynomials

Synonym

Bernstein basis functions

Generated on Sat Feb 10 12:35:26 2018 by

LaTeXML
properties of Bernstein polynomial

properties of Bernstein polynomial

The

Bernstein polynomials

B

i

n

⁢

(

t

)

have the following properties:

0.1

Non negativity

The

polynomials

are non-negative over the

interval

[

0

,

1

]

.

B

i

n

⁢

(

t

)

≥

0

0

≤

t

≤

1

0.2

Symmetry

The set of polynomials of degree

n

is

symmetric

with respect to

t

=

1

/

2

.

B

i

n

⁢

(

t

)

=

B

n

-

i

n

⁢

(

1

-

t

)

0.3

Maximum

Each polynomial has only one maximum over the interval

[

0

,

1

]

at

t

=

i

n

.

0.4

Normalization

The set of polynomials of degree

n

forms a partition of unity.

∑

i

=

0

n

B

i

n

⁢

(

t

)

=

1

0.5

Degree raising

A polynomial can always be written as a

linear combination

of polynomials of higher degree.

B

i

n

-

1

⁢

(

t

)

=

n

-

i

n

⁢

B

i

n

⁢

(

t

)

+

i

+

1

n

⁢

B

i

+

1

n

⁢

(

t

)

Title

properties of Bernstein polynomial

Canonical name

PropertiesOfBernsteinPolynomial

Date of creation

2013-03-22 17:24:13

Last modified on

2013-03-22 17:24:13

Owner

stitch (17269)

Last modified by

stitch (17269)

Numerical id

10

Author

stitch (17269)

Entry type

Derivation

Classification

msc 65D17

Generated on Sat Feb 10 12:35:28 2018 by

LaTeXML
code for Simpson’s rule

code for Simpson’s rule

Python code for Simpson’s rule

\PMlinkescapetext{
from math import *
def f(x):
#function to integrate
return sin(x)

def simpson_rule(a,b):
#Approximation by Simpson's rule
c=(a+b)/2.0
h=abs(b-a)/2.0
return h*(f(a)+4.0*f(c)+f(b))/3.0

# Calculates integral of f(x) from 0 to 1
print simpson_rule(0,1)
}

Integrating

sin

⁡

x

from

0

to

1

with the previous code gives

0.45986218971

⁢

…

whereas the true value is

1

-

cos

⁡

1

=

0.459697694131860282599063392557

⁢

…

.

Title

code for Simpson’s rule

Canonical name

CodeForSimpsonsRule

Date of creation

2013-03-22 14:50:52

Last modified on

2013-03-22 14:50:52

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Algorithm

Classification

msc 65D32

Related topic

NewtonAndCotesFormulas

Generated on Sat Feb 10 12:35:30 2018 by

LaTeXML
Newton and Cotes formulas

Newton and Cotes formulas

The usual way of numerically integrate a function, is to find a simpler function which approximates the given function and then integrating the

interpolation

function.
That is, if we want to find

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

, we find an approximating function

p

⁢

(

x

)

such that

f

⁢

(

x

)

and

p

⁢

(

x

)

be close (on some concept of distance) and then we say

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

≈

∫

a

b

p

⁢

(

x

)

⁢

𝑑

x

The simplest

approximation

functions are

polynomials

. If we evaluate

f

⁢

(

x

)

at some points

x

0

,

x

1

,

…

,

x

n

, we can use Lagrange’s interpolating polynomial to find a polynomial

p

⁢

(

x

)

with degree

n

such that

p

⁢

(

x

j

)

=

f

⁢

(

x

j

)

for

j

=

0

,

1

,

…

,

n

.

Newton and Cotes’

integration formulas

are obtained when the

x

0

,

x

1

,

…

,

x

n

are sampled evenly over the

interval

, and then

Lagrange interpolating polynomials

are used to approximate the function.

The

Newton and Cotes formulas

for small values of

n

are given on the following table.

n

∫

p

⁢

(

x

)

Name

1

h

2

⁢

(

f

⁢

(

x

0

)

+

f

⁢

(

x

1

)

)

Trapezoidal rule

2

h

3

⁢

(

f

⁢

(

x

0

)

+

4

⁢

f

⁢

(

x

1

)

+

f

⁢

(

x

2

)

)

Simpson’s rule

3

3

⁢

h

8

⁢

(

f

⁢

(

x

0

)

+

3

⁢

f

⁢

(

x

1

)

+

3

⁢

f

⁢

(

x

3

)

+

f

⁢

(

x

3

)

)

Simpson’s 3/8 rule

4

2

⁢

h

45

⁢

(

7

⁢

f

⁢

(

x

0

)

+

32

⁢

f

⁢

(

x

1

)

+

12

⁢

f

⁢

(

x

2

)

+

32

⁢

f

⁢

(

x

3

)

+

7

⁢

f

⁢

(

x

4

)

)

Milne’s rule

recalling that

x

0

,

x

1

,

…

,

x

n

are evenly spaced on

[

a

,

b

]

.

Since the Simpson’s rule is actually the Newton and Cotes formula for

n

=

2

, the proof of Simpson’s rule illustrates this method.

Title

Newton and Cotes formulas

Canonical name

NewtonAndCotesFormulas

Date of creation

2013-03-22 14:50:28

Last modified on

2013-03-22 14:50:28

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 65D32

Synonym

Newton-Cotes

Related topic

SimpsonsRule

Related topic

CodeForSimpsonsRule

Generated on Sat Feb 10 12:35:32 2018 by

LaTeXML
proof of Simpson’s rule

proof of Simpson’s rule

We want to derive Simpson’s rule for

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

.

We will use

Newton and Cotes formulas

for

n

=

2

. In this case,

x

0

=

a

,

x

2

=

b

and

x

1

=

(

a

+

b

)

/

2

. We use Lagrange’s interpolation formula to get a

polynomial

p

⁢

(

x

)

such that

p

⁢

(

x

j

)

=

f

⁢

(

x

j

)

for

j

=

0

,

1

,

2

.

The corresponding interpolating polynomial is

p

⁢

(

x

)

=

f

⁢

(

x

1

)

⁢

(

x

-

x

2

)

⁢

(

x

-

x

3

)

(

x

1

-

x

2

)

⁢

(

x

1

-

x

3

)

+

f

⁢

(

x

2

)

⁢

(

x

-

x

1

)

⁢

(

x

-

x

3

)

(

x

2

-

x

1

)

⁢

(

x

2

-

x

3

)

+

f

⁢

(

x

3

)

⁢

(

x

-

x

1

)

⁢

(

x

-

x

2

)

(

x

3

-

x

1

)

⁢

(

x

3

-

x

2

)

.

and thus

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

≈

∫

a

b

f

⁢

(

x

1

)

⁢

(

x

-

x

2

)

⁢

(

x

-

x

3

)

(

x

1

-

x

2

)

⁢

(

x

1

-

x

3

)

+

f

⁢

(

x

2

)

⁢

(

x

-

x

1

)

⁢

(

x

-

x

3

)

(

x

2

-

x

1

)

⁢

(

x

2

-

x

3

)

+

f

⁢

(

x

3

)

⁢

(

x

-

x

1

)

⁢

(

x

-

x

2

)

(

x

3

-

x

1

)

⁢

(

x

3

-

x

2

)

⁢

d

⁢

x

.

Since integration is linear, we are concerned only with integrating each term in the sum. Now, taking

x

j

=

a

+

h

⁢

j

where

j

=

0

,

1

,

2

and

h

=

|

b

-

a

|

/

2

, we can rewrite the

quotients

on the last integral as

∫

a

b

p

⁢

(

x

)

⁢

𝑑

x

=

h

⁢

f

⁢

(

x

0

)

⁢

∫

0

2

(

t

-

1

)

⁢

(

t

-

2

)

(

0

-

1

)

⁢

(

0

-

2

)

⁢

𝑑

t

+

h

⁢

f

⁢

(

x

1

)

⁢

∫

0

2

(

t

-

0

)

⁢

(

t

-

2

)

(

1

-

0

)

⁢

(

1

-

2

)

⁢

𝑑

t

+

h

⁢

f

⁢

(

x

2

)

⁢

∫

0

2

(

t

-

0

)

⁢

(

t

-

1

)

(

2

-

0

)

⁢

(

2

-

1

)

⁢

𝑑

t

.

and if we calculate the integrals on the last

expression

we get

∫

a

b

p

⁢

(

x

)

⁢

𝑑

x

=

h

⁢

f

⁢

(

x

0

)

⁢

1

3

+

h

⁢

f

⁢

(

x

1

)

⁢

4

3

+

h

⁢

f

⁢

(

x

2

)

⁢

1

3

,

which is Simpson’s rule:

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

≈

h

3

⁢

(

f

⁢

(

x

0

)

+

4

⁢

f

⁢

(

x

1

)

+

f

⁢

(

x

2

)

)

.

Title

proof of Simpson’s rule

Canonical

name

ProofOfSimpsonsRule

Date of creation

2013-03-22 14:50:25

Last modified on

2013-03-22 14:50:25

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Proof

Classification

msc 65D32

Classification

msc 41A55

Classification

msc 26A06

Classification

msc 28-00

Generated on Sat Feb 10 12:35:34 2018 by

LaTeXML
eigenvector

eigenvector

Let

A

be an

n

×

n

square matrix

and

x

an

n

×

1

column vector

. Then a (right)

eigenvector

of

A

is a nonzero vector

x

such that

A

⁢

x

=

λ

⁢

x

for some scalar

λ

, i.e. such that the image of

x

under the transformation

A

is a

scalar

of

x

. One can similarly define

left eigenvectors

in the case that

A

acts on the right.

One can find eigenvectors by first

finding eigenvalues

, then for each

eigenvalue

λ

i

, solving the system

(

A

-

λ

i

⁢

I

)

⁢

x

i

=

0

to find a form which characterizes the eigenvector

x

i

(any of

x

i

is also an eigenvector). Of course, this is not necessarily the best way to do it; for this, see

singular value decomposition

.

Title

eigenvector

Canonical name

Eigenvector

Date of creation

2013-03-22 12:11:55

Last modified on

2013-03-22 12:11:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 65F15

Classification

msc 65-00

Classification

msc 15A18

Classification

msc 15-00

Related topic

SingularValueDecomposition

Related topic

Eigenvalue

Related topic

EigenvalueProblem

Related topic

SimilarMatrix

Related topic

DiagonalizationLinearAlgebra

Defines

scalar multiple

Generated on Sat Feb 10 12:35:39 2018 by

LaTeXML
example of Gram-Schmidt orthogonalization

example of Gram-Schmidt orthogonalization

Let us work with the standard

inner product

on

ℝ

3

(

dot product

) so we can get a nice geometrical visualization.

Consider the three vectors

v

1

=

(

3

,

0

,

4

)

v

2

=

(

-

6

,

-

4

,

1

)

v

3

=

(

5

,

0

,

-

3

)

which are

linearly independent

(the

determinant

of the matrix

A

=

(

v

1

|

v

2

|

v

3

)

=

116

≠

0

)

but are not

orthogonal

.

We will now apply

Gram-Schmidt

to get three vectors

w

1

,

w

2

,

w

3

which span the same

subspace

(in this case, all

R

3

) and orthogonal to each other.

First we take

w

1

=

v

1

=

(

3

,

0

,

4

)

. Now,

w

2

=

v

2

-

w

1

⋅

v

2

∥

w

1

∥

2

⁢

w

1

that is,

w

2

=

(

-

108

25

,

-

4

,

81

25

)

and finally

w

3

=

v

3

-

w

1

⋅

v

3

∥

w

1

∥

2

⁢

w

1

-

w

2

⋅

v

3

∥

w

2

∥

2

⁢

w

2

which gives

w

3

=

(

1856

1129

,

3132

1129

,

1392

1129

)

and so

{

w

1

,

w

2

,

w

3

}

is an

orthogonal set

of vectors such that

⟨

w

1

,

w

2

,

w

3

⟩

=

⟨

v

1

,

v

2

,

v

3

⟩

.

If we rather consider

{

w

1

/

∥

w

1

∥

,

w

2

/

∥

w

2

∥

,

w

3

/

∥

w

3

∥

}

then we get an

orthonormal set

.

Title

example of

Gram-Schmidt orthogonalization

Canonical name

ExampleOfGramSchmidtOrthogonalization

Date of creation

2013-03-22 15:03:02

Last modified on

2013-03-22 15:03:02

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Example

Classification

msc 65F25

Related topic

ProofOfGramSchmidtOrthogonalizationProcedure

Generated on Sat Feb 10 12:35:53 2018 by

LaTeXML