aarjaneiro/planetmath huggingface dataset:
Gram-Schmidt orthogonalization

Gram-Schmidt orthogonalization

Any set of

linearly independent

vectors

v

1

,

…

,

v

n

can be converted into a set of

orthogonal vectors

q

1

,

…

,

q

n

by the

Gram-Schmidt process

. In three

dimensions

,

v

1

determines a line; the vectors

v

1

and

v

2

determine a plane. The vector

q

1

is the

unit vector

in the

direction

v

1

. The (unit) vector

q

2

lies in the plane of

v

1

,

v

2

, and is normal to

v

1

(on the same side as

v

2

. The (unit) vector

q

3

is normal to the plane of

v

1

,

v

2

, on the same side as

v

3

, etc.

In general, first set

u

1

=

v

1

, and then each

u

i

is made

orthogonal

to the preceding

u

1

,

…

⁢

u

i

-

1

by

subtraction

of the

projections

of

v

i

in the directions of

u

1

,

…

,

u

i

-

1

:

u

i

=

v

i

-

∑

j

=

1

i

-

1

u

j

T

⁢

v

i

u

j

T

⁢

u

j

⁢

u

j

The

i

vectors

u

i

span the same

subspace

as the

v

i

. The vectors

q

i

=

u

i

/

||

u

i

||

are

orthonormal

. This leads to the following theorem:

Theorem.

Any

m

×

n

matrix

A

with linearly independent columns can be factorized into a

product

,

A

=

Q

⁢

R

. The columns of

Q

are orthonormal and

R

is

upper triangular

and

invertible

.

This “classical”

Gram-Schmidt

method is often numerically unstable, see [Golub89] for a “modified” Gram-Schmidt method.

References

•

Originally from The Data Analysis Briefbook
(

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

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

Golub89

Gene H. Golub and Charles F. van Loan: Matrix Computations, 2nd edn., The John Hopkins University Press, 1989.

Title

Gram-Schmidt orthogonalization

Canonical name

GramSchmidtOrthogonalization

Date of creation

2013-03-22 12:06:14

Last modified on

2013-03-22 12:06:14

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Algorithm

Classification

msc 65F25

Synonym

Gram-Schmidt decomposition

Synonym

Gram-Schmidt orthonormalization

Synonym

Gram-Schmidt process

Related topic

HouseholderTransformation

Related topic

GivensRotation

Related topic

QRDecomposition

Related topic

AnExampleForSchurDecomposition

Generated on Sat Feb 10 12:35:55 2018 by

LaTeXML
orthonormal set

orthonormal set

Definition

An

orthonormal set

is a subset

S

of an

inner product space

,
such that

⟨

x

,

y

⟩

=

δ

x

⁢

y

for all

x

,

y

∈

S

.
Here

⟨

⋅

,

⋅

⟩

is the

inner product

,
and

δ

is the

Kronecker delta

.

More verbosely, we may say that an orthonormal set
is a subset

S

of an inner product space
such that the following two conditions hold:

1.

If

x

,

y

∈

S

and

x

≠

y

, then

x

is

orthogonal

(

http://planetmath.org/OrthogonalVector

) to

y

.

2.

If

x

∈

S

, then the norm of

x

is

1

.

Stated this way, the origin of the term is clear:
an orthonormal set of vectors is both orthogonal and normalized.

Notes

Note that the

empty set

is orthonormal,
as is a set consisting of a single vector of unit norm
in an inner product space.

The columns (or rows) of a real

orthogonal matrix

form an orthonormal set.
In fact, this is an example of an

orthonormal basis

.

Applications

A standard application is finding an orthonormal basis for a

vector space

,
such as by

Gram-Schmidt orthonormalization

.
Orthonormal bases are computationally simple to work with.

Title

orthonormal set

Canonical name

OrthonormalSet

Date of creation

2013-03-22 12:07:24

Last modified on

2013-03-22 12:07:24

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

14

Author

yark (2760)

Entry type

Definition

Classification

msc 65F25

Related topic

OrthogonalPolynomials

Related topic

OrthonormalBasis

Defines

orthonormal

Generated on Sat Feb 10 12:35:57 2018 by

LaTeXML
QR decomposition

QR decomposition

1

QR Decomposition

Orthogonal matrix

triangularization (

QR decomposition

) reduces a real

m

×

n

matrix

A

with

m

≥

n

and full rank to a much simpler form. It guarantees

numerical stability

by minimizing errors caused by machine roundoffs. A suitably chosen orthogonal matrix

Q

will triangularize the given matrix:

A

=

Q

⁢

[

R

0

]

with the

n

×

n

right triangular matrix

R

. One only has then to solve the triangular system

R

⁢

x

=

P

⁢

b

, where

P

consists of the first

n

rows of

Q

.

The least squares problem

A

⁢

x

≈

b

is easy to solve with

A

=

Q

⁢

R

and

Q

an orthogonal matrix (here and henceforth

R

is the entire

m

×

n

augmented matrix from above). The solution

x

=

(

A

T

⁢

A

)

-

1

⁢

A

T

⁢

b

becomes

x

=

(

R

T

⁢

Q

T

⁢

Q

⁢

R

)

-

1

⁢

R

T

⁢

Q

T

⁢

b

=

(

R

T

⁢

R

)

-

1

⁢

R

T

⁢

Q

T

⁢

b

=

R

-

1

⁢

Q

T

⁢

b

This is a matrix-vector

multiplication

Q

T

⁢

b

, followed by the solution of the triangular system

R

⁢

x

=

Q

T

⁢

b

by back-substitution. The QR factorization saves us the formation of

A

T

⁢

A

and the solution of the

normal equations

.

Many different methods exist for the QR decomposition, e.g. the

Householder transformation

, the

Givens rotation

, or the

Gram-Schmidt decomposition

.

References

1

The Data Analysis Briefbook.

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

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

Title

QR decomposition

Canonical

name

QRDecomposition

Date of creation

2013-03-22 12:06:04

Last modified on

2013-03-22 12:06:04

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Definition

Classification

msc 65F25

Classification

msc 15A23

Synonym

QR factorization

Synonym

QR-decomposition

Synonym

QR-factorization

Related topic

LUDecomposition

Related topic

GramSchmidtOrthogonalization

Generated on Sat Feb 10 12:36:03 2018 by

LaTeXML
Frobenius matrix norm

Frobenius matrix norm

Let

R

be a ring with a valuation

|

⋅

|

and let

M

⁢

(

R

)

denote the set of matrices over

R

. The

Frobenius norm

function

or

Euclidean matrix norm

is the norm function

|

|

⋅

|

|

F

:

M

(

R

)

→

ℝ

given by

||

A

||

F

=

∑

i

=

1

m

∑

j

=

1

n

|

a

i

⁢

j

|

2

,

where

m

and

n

respectively denote the number of rows and columns of

A

. Note

A

need not be square for this definition. A more concise (though ) definition, in the case that

R

=

ℝ

or

ℂ

, is

||

A

||

F

=

trace

⁢

(

A

*

⁢

A

)

,

where

A

*

denotes the

conjugate transpose

of

A

.

Some :

•

Denote the columns of

A

by

A

i

. A nice property of the norm is that

||

A

||

F

2

=

||

A

1

||

2

2

+

||

A

2

||

2

2

+

⋯

+

||

A

n

||

2

2

.

•

Let

A

be a

square matrix

and let

U

be a

unitary matrix

of same size as

A

. Then

||

A

||

F

=

||

U

∗

⁢

A

⁢

U

||

F

where

U

∗

is the conjugate transpose of

U

.

•

If

A

⁢

B

is defined, then

||

A

⁢

B

||

F

≤

||

A

||

F

⁢

||

B

||

F

.

Title

Frobenius matrix norm

Canonical name

FrobeniusMatrixNorm

Date of creation

2013-03-22 11:43:25

Last modified on

2013-03-22 11:43:25

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

25

Author

mathcam (2727)

Entry type

Definition

Classification

msc 65F35

Classification

msc 15A60

Classification

msc 18-00

Synonym

Euclidean matrix norm

Synonym

matrix F-norm

Synonym

Hilbert-Schmidt norm

Related topic

MatrixNorm

Related topic

MatrixPnorm

Related topic

VectorNorm

Related topic

VectorPnorm

Related topic

ShursInequality

Related topic

trace

Related topic

transpose

Related topic

Transpose

Related topic

MatrixLogarithm

Related topic

FrobeniusProduct

Generated on Sat Feb 10 12:36:05 2018 by

LaTeXML
matrix condition number is greater or equal to 1

matrix condition number is greater or equal to

1

Theorem 1

.

The

matrix condition number

is greater or equal

1

.

Proof.

Let

A

be a

square matrix

. Then by

properties

of a

matrix norm

,

1

=

||

I

||

=

||

A

⁢

A

-

1

||

≤

||

A

-

1

||

⁢

||

A

||

.

∎

Title

matrix condition number is greater or equal to

1

Canonical

name

MatrixConditionNumberIsGreaterOrEqualTo1

Date of creation

2013-03-22 15:28:15

Last modified on

2013-03-22 15:28:15

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

10

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 65F35

Classification

msc 15A12

Related topic

MatrixConditionNumber

Generated on Sat Feb 10 12:36:07 2018 by

LaTeXML
matrix norm

matrix norm

A is a norm on the set

M

m

×

n

⁢

(

R

)

of

m

×

n

matrices over a ring

R

, and is
usually denoted by

|

|

⋅

|

|

.

The

Frobenius matrix norm

|

|

⋅

|

|

F

and the matrix

p

-norm (

http://planetmath.org/MatrixPNorm

)

|

|

⋅

|

|

p

are the most frequently used

matrix norms

.

A

vector norm

is a matrix norm for which the ring

R

is a field.

Title

matrix norm

Canonical

name

MatrixNorm

Date of creation

2013-03-22 11:43:20

Last modified on

2013-03-22 11:43:20

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

17

Author

mathcam (2727)

Entry type

Definition

Classification

msc 65F35

Classification

msc 43-02

Classification

msc 00A69

Related topic

VectorNorm

Related topic

FrobeniusMatrixNorm

Related topic

MatrixPnorm

Generated on Sat Feb 10 12:36:09 2018 by

LaTeXML
Pascal matrix

Pascal matrix

Definition

The

Pascal matrix

P

of order

n

is the real
square

n

×

n

matrix whose entries are

[

1

]

P

i

⁢

j

=

(

i

+

j

-

2

j

-

1

)

.

For

n

=

5

,

P

=

(

1

1

1

1

1

1

2

3

4

5

1

3

6

10

15

1

4

10

20

35

1

5

15

35

70

)

,

so we see that the Pascal matrix contains the

Pascal triangle

on its antidiagonals.

Pascal matrices are

ill-conditioned

. However, the

inverse

of the

n

×

n

Pascal matrix is known explicitly and given in

[

1

]

.
The

characteristic polynomial

of a Pascal triangle is a

reciprocal
polynomial

[

1

]

.

References

1

N.J. Higham,

Accuracy and Stability of Numerical Algorithms

,
2nd ed., SIAM, 2002.

Title

Pascal matrix

Canonical name

PascalMatrix

Date of creation

2013-03-22 13:44:54

Last modified on

2013-03-22 13:44:54

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Definition

Classification

msc 65F35

Classification

msc 15A12

Classification

msc 15A09

Classification

msc 15A57

Generated on Sat Feb 10 12:36:11 2018 by

LaTeXML
pivoting

pivoting

1

Pivoting

Pivoting

is a process performed on a matrix in order to improve

numerical stability

.

Partial pivoting

of an

n

×

n

matrix is the

sorting

of the rows of the matrix so that row

i

contains the maximum absolute column value for column

i

, among all rows

i

,

…

,

n

. That is, we begin by swapping row 1 with the row that has the largest

absolute value

for the first column, then swap row 2 with the row that has the largest magnitude for the second column (among rows 2 and below), and so on.

Complete pivoting

is a reordering of both rows and columns, using the same method as above. It is usually not necessary to ensure numerical stability.

Pivoting can be represented as

multiplication

by

permutation matrices

.

References

1

G. H. Golub, C. F. Loan, Matrix

Computations

, 3rd edition, Johns Hopkins, 1996.

Title

pivoting

Canonical

name

Pivoting

Date of creation

2013-03-22 12:06:54

Last modified on

2013-03-22 12:06:54

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

8

Author

akrowne (2)

Entry type

Algorithm

Classification

msc 65F35

Classification

msc 65F30

Synonym

total pivoting

Synonym

complete pivoting

Synonym

partial pivoting

Generated on Sat Feb 10 12:36:12 2018 by

LaTeXML
singular

singular

1

Singular

An

m

×

n

matrix

A

with entries from a field is called

singular

if its rows or columns are

linearly dependent

. This is

equivalent

to the following conditions:

1.

The

nullity

of

A

is

greater than zero

(

null

⁡

(

A

)

>

0

).

2.

The

homogeneous

linear system

A

⁢

𝐱

=

0

has a non-trivial solution.

If

m

=

n

this is equivalent to the following conditions:

1.

The

determinant

det

⁡

(

A

)

=

0

.

2.

The rank of

A

is less than

n

.

Title

singular

Canonical

name

Singular

Date of creation

2013-03-22 11:57:38

Last modified on

2013-03-22 11:57:38

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 65F35

Classification

msc 15A12

Synonym

non-invertible

Synonym

singular transformation

Generated on Sat Feb 10 12:36:14 2018 by

LaTeXML
Rayleigh-Ritz method

Rayleigh-Ritz method

The Rayleigh-Ritz method is an algorithm for obtaining approximate solutions to eigenvalue ODEs. It can be neatly summarized as follows:

1.

Choose an approximate form for the

eigenfunction

with the lowest eigenvalue (the ground state wavefunction, in the language of quantum mechanics). Include one or more free

parameters

.

2.

Find the expectation value of the eigenvalue with respect to the trial eigenfunction.

3.

Minimize the resulting equation with respect to the free parameter(s), hence finding a value for the free parameter.

4.

Substitute this new eigenfunction back into the expectation value.

5.

The expectation value obtained is an

upper bound

for the actual eigenvalue of the true eigenfunction.

1

Example

Consider the Schrödinger equation for a one-dimensional

harmonic oscillator

potential

:

(

-

ℏ

2

2

⁢

m

⁢

∂

2

∂

⁡

x

2

+

1

2

⁢

m

2

⁢

ω

2

)

⁢

ψ

=

E

⁢

ψ

where

m

is the mass of the particle in the well, and

ω

is the angular velocity a classical particle would move with in the well. This equation can be solved exactly using Frobenius’ method, and leads to eigenfunctions of the form of

Hermite polynomials

multiplied by

Gaussians

, and half-integer eigenvalues of the form

E

n

=

(

n

+

1

/

2

)

⁢

ℏ

⁢

ω

. Since the solutions are known, it is a good test case. We choose the ground state wavefunction of the infinite potential well as our trial eigenfunction:

ψ

=

cos

⁡

(

π

⁢

x

2

⁢

a

)

a

with

a

as our free parameter.
We now find the expectation value:

⟨

E

⟩

=

⟨

ψ

|

H

^

|

ψ

⟩

=

∫

-

a

a

ψ

*

⁢

H

^

⁢

ψ

⁢

𝑑

x

Evaluating the

integral

, we find

⟨

E

⟩

=

ℏ

2

⁢

π

2

8

⁢

m

⁢

a

2

+

m

⁢

ω

2

⁢

a

2

⁢

(

1

6

-

1

p

⁢

i

2

)

We now minimise this with respect to

a

to obtain:

2

⁢

m

⁢

ω

2

⁢

a

⁢

(

1

6

-

1

π

2

)

=

ℏ

2

⁢

π

2

4

⁢

m

⁢

a

2

Hence:

a

=

π

⁢

(

3

4

⁢

(

π

2

-

6

)

)

1

4

⁢

(

ℏ

m

⁢

ω

)

1

2

Substituting this into the expecation value

⟨

E

⟩

we obtain

⟨

E

⟩

=

1

2

⁢

(

π

2

-

6

3

)

1

2

⁢

ℏ

⁢

ω

⟨

E

⟩

≈

0.568

⁢

ℏ

⁢

ω

The analytical value is of course

0.5

⁢

ℏ

⁢

ω

. Considering the crudeness of the

approximation

used, the result is impressive.

Title

Rayleigh-Ritz method

Canonical

name

RayleighRitzMethod

Date of creation

2013-03-22 17:52:18

Last modified on

2013-03-22 17:52:18

Owner

invisiblerhino (19637)

Last modified by

invisiblerhino (19637)

Numerical id

8

Author

invisiblerhino (19637)

Entry type

Definition

Classification

msc 65L60

Generated on Sat Feb 10 12:36:31 2018 by

LaTeXML
integral transform

integral transform

A generic

integral transform

takes the form

F

⁢

(

p

)

=

∫

α

β

K

⁢

(

p

,

t

)

⁢

f

⁢

(

t

)

⁢

𝑑

t

,

with

p

being the

transform parameter

.

Note that the

transform

takes a

function

f

⁢

(

t

)

and produces a new function

F

⁢

(

p

)

.

The function

K

⁢

(

p

,

t

)

is called the

kernel

of the transform. The kernel of an integral transform, along with the limits (

http://planetmath.org/DefiniteIntegral

)

α

and

β

, distinguish a particular integral transform from another.

Examples

•

Laplace transform

α

=

0

,

β

=

∞

,

K

⁢

(

p

,

t

)

=

e

-

p

⁢

t

,

F

⁢

(

p

)

=

∫

0

∞

e

-

p

⁢

t

⁢

f

⁢

(

t

)

⁢

𝑑

t

.

•

Laplace-Carson transform

α

=

0

,

β

=

∞

,

K

⁢

(

p

,

t

)

=

p

⁢

e

-

p

⁢

t

,

F

⁢

(

p

)

=

∫

0

∞

p

⁢

e

-

p

⁢

t

⁢

f

⁢

(

t

)

⁢

𝑑

t

.

•

Fourier transform

α

=

-

∞

,

β

=

∞

,

K

⁢

(

p

,

t

)

=

1

2

⁢

π

⁢

e

-

i

⁢

p

⁢

t

,

F

⁢

(

p

)

=

1

2

⁢

π

⁢

∫

-

∞

∞

e

-

i

⁢

p

⁢

t

⁢

f

⁢

(

t

)

⁢

𝑑

t

.

Title

integral transform

Canonical name

IntegralTransform

Date of creation

2013-03-22 12:34:03

Last modified on

2013-03-22 12:34:03

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

10

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 65R10

Related topic

ContourIntegral

Related topic

GroupHomomorphism

Defines

kernel

Defines

transform parameter

Generated on Sat Feb 10 12:36:41 2018 by

LaTeXML
Vandermonde matrix

Vandermonde matrix

A

Vandermonde matrix

is any

(

n

+

1

)

×

(

n

+

1

)

matrix of the form

[

1

x

0

x

0

2

⋯

x

0

n

1

x

1

x

1

2

⋯

x

1

n

⋮

⋮

⋮

⋱

⋮

1

x

n

x

n

2

⋯

x

n

n

]

Vandermonde matrices usually arise when considering systems of polynomials evaluated at specific points (i.e. in

interpolation

or

approximation

). This may happen, for example, when trying to solve for constants from

initial conditions

in systems of differential equations or

recurrence relations

.

Vandermonde matrices also appear in the computation of FFTs (Fast Fourier Transforms). Here the fact that Vandermonde systems

V

⁢

z

=

b

can be solved in

𝒪

⁢

(

n

⁢

log

⁡

n

)

flops by taking advantage of their comes into play.

0.1

References

1.

Golub and Van Loan,

Matrix Computations

, Johns Hopkins University Press 1993

Title

Vandermonde matrix

Canonical

name

VandermondeMatrix

Date of creation

2013-03-22 13:04:19

Last modified on

2013-03-22 13:04:19

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

4

Author

akrowne (2)

Entry type

Definition

Classification

msc 65T50

Classification

msc 65F99

Classification

msc 15A57

Generated on Sat Feb 10 12:36:52 2018 by

LaTeXML
wavelet

wavelet

Motivation

Wavelets

can be used to analyze

functions

in

L

2

⁢

(

ℝ

)

(the space of all Lebesgue absolutely

square integrable

functions defined on the

real numbers

to the

complex numbers

) in much the same way the

complex

exponentials

are used in the

Fourier transform

, but wavelets offer the advantage of not only describing the frequency content of a function, but also providing information on the time localization of that frequency content.

Definition

A (more properly, an

orthonormal dyadic

)

wavelet

is a function

ψ

⁢

(

t

)

∈

L

2

⁢

(

ℝ

)

such that the family of functions

ψ

j

⁢

k

≡

2

j

/

2

⁢

ψ

⁢

(

2

j

⁢

t

-

k

)

,

where

j

,

k

∈

ℤ

, is an

orthonormal basis

in the

Hilbert space

L

2

⁢

(

ℝ

)

.

Notes

The scaling factor of

2

j

/

2

ensures that

∥

ψ

j

⁢

k

∥

=

∥

ψ

∥

=

1

. These type of wavelets (the most popular), are known as dyadic wavelets because the scaling factor is a power of 2. It is not obvious from the definition that wavelets even exist, or how to construct one; the Haar wavelet is the standard example of a wavelet, and one technique used to construct wavelets. Generally, wavelets are constructed from a

multiresolution analysis

, but they can also be generated using

wavelet sets

.

Title

wavelet

Canonical name

Wavelet

Date of creation

2013-03-22 14:26:41

Last modified on

2013-03-22 14:26:41

Owner

swiftset (1337)

Last modified by

swiftset (1337)

Numerical id

11

Author

swiftset (1337)

Entry type

Definition

Classification

msc 65T60

Classification

msc 46C99

Related topic

FourierTransform

Related topic

MultiresolutionAnalysis

Related topic

WaveletSet2

Defines

wavelet

Defines

orthonormal dyadic wavelet

Generated on Sat Feb 10 12:36:54 2018 by

LaTeXML
Dynamic Stream

Dynamic Stream

A

dynamic stream

can represent an information flow — a sequence of messages
encoded by the sender and communicated using a transmission channel
possibly distorted with noise, to a receiver whose goal is to reconstruct the
sender’s messages and interpret message semantics.
—
From:
”SHANNON, C. E. 1948. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656.”

Title

Dynamic Stream

Canonical name

DynamicStream

Date of creation

2013-03-22 15:31:34

Last modified on

2013-03-22 15:31:34

Owner

zshao (11001)

Last modified by

zshao (11001)

Numerical id

4

Author

zshao (11001)

Entry type

Definition

Classification

msc 68-00

Related topic

Information

Generated on Sat Feb 10 12:37:08 2018 by

LaTeXML
XML

XML

Extensible Markup Language (XML) is a flexible, text based, format derived from SGML. XML was originally designed for satisfying electronic publishing work. XML is a standard for data exchange on the Web and Internet.
XML and SGML can be used for describing the structure of metadata.
XML documents are made up of storage units called entities, which contain either parsed or unparsed data. Parsed data consists of character data or markup. Markup encodes a description of the document’s storage layout and logical structure. XML provides a mechanism to impose constraints on the storage layout and logical structure.

The are published and maintained by the

http://www.w3.org/XML/

W3C, they are and freely available on their website.

Title

XML

Canonical

name

XML

Date of creation

2013-03-22 15:31:43

Last modified on

2013-03-22 15:31:43

Owner

yota (10184)

Last modified by

yota (10184)

Numerical id

5

Author

yota (10184)

Entry type

Definition

Classification

msc 68-00

Generated on Sat Feb 10 12:37:09 2018 by

LaTeXML
OpenPFGW

OpenPFGW

OpenPFGW

, formerly called

PrimeForm

, is an open source

primality testing

program on the Intel platform that has found many of the large primes listed in the Prime Pages. It uses sieve methods, pseudorandom permutation (PRP) as well as methods for specific kinds of primes, as programmed by George Woltman for GIMPS. As of 2006, the largest prime it has found is

19

×

2

2206266

+

1

.

According to the Prime Pages, the official website is

http://tech.groups.yahoo.com/group/primeform/

a Yahoo! Group.

Title

OpenPFGW

Canonical name

OpenPFGW

Date of creation

2013-03-22 16:36:59

Last modified on

2013-03-22 16:36:59

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Topic

Classification

msc 68-04

Synonym

PrimeForm

Generated on Sat Feb 10 12:37:11 2018 by

LaTeXML
Amdahl’s Law

Amdahl’s Law

Amdahl’s Law reveals the maximum

speedup

that can be expected from parallel algorithms given the proportion of parts that must be computed sequentially. It gives the speedup

S

as

S

≤

1

f

+

(

1

-

f

)

/

N

Where

f

is the fraction of the problem that must be computed sequentially and

N

is the number of processors.

Note that as

f

approaches zero,

S

nears

N

, which we’d expect from a perfectly parallelizable

algorithm

.

Title

Amdahl’s Law

Canonical name

AmdahlsLaw

Date of creation

2013-03-22 12:04:15

Last modified on

2013-03-22 12:04:15

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

17

Author

alozano (2414)

Entry type

Theorem

Classification

msc 68M20

Generated on Sat Feb 10 12:37:12 2018 by

LaTeXML
efficiency

efficiency

Efficiency

is a measure of parallel algorithm performance per processor. The efficiency

ϕ

is defined as

ϕ

=

S

N

where

S

is the

speedup

associated with the

algorithm

and

N

is the number of processors.

Title

efficiency

Canonical name

Efficiency

Date of creation

2013-03-22 12:04:17

Last modified on

2013-03-22 12:04:17

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

6

Author

akrowne (2)

Entry type

Definition

Classification

msc 68M20

Generated on Sat Feb 10 12:37:14 2018 by

LaTeXML