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### Transcript of Preconditioners for Singular Black Box for Singular Black Box Matrices ... qcc cccc cc ccc cc c cc c...

• Preconditioners for Singular Black Box Matrices

William J. Turner

Department of Mathematics & Computer ScienceWabash College

Crawfordsville, IN 47933

ISSAC 2005 : Beijing, China27 July 2005

• Outline

Generalized NetworksArbitrary Radix Switching NetworksGeneric Exchange Matrices

Rank Preconditioner

Generalize butterfly switch to radix- switch:

Example (Radix- switches: = 2, 3, 4)

x1

AAAA

AAAA

A x2

}}}}

}}}}

}

x1 x2

x1

AAAA

AAAA

A

PPPPPP

PPPPPP

PPx2

}}}}

}}}}

}

AAAA

AAAA

A x3

nnnnnn

nnnnnn

nn

}}}}

}}}}

}

x1 x2 x3

x1

AAAA

AAAA

A

PPPPPP

PPPPPP

PP

UUUUUUUU

UUUUUUUU

UUUUx2

}}}}

}}}}

}

AAAA

AAAA

A

PPPPPP

PPPPPP

PPx3

nnnnnn

nnnnnn

nn

}}}}

}}}}

}

AAAA

AAAA

A x4

iiiiiiii

iiiiiiii

iiii

nnnnnn

nnnnnn

nn

}}}}

}}}}

}

x1 x2 x3 x4

Generalize butterfly switch to radix- switch:

Example (Radix- switches: = 2, 3, 4)

x1

AAAA

AAAA

A x2

}}}}

}}}}

}

x1 x2

x1

AAAA

AAAA

A

PPPPPP

PPPPPP

PPx2

}}}}

}}}}

}

AAAA

AAAA

A x3

nnnnnn

nnnnnn

nn

}}}}

}}}}

}

x1 x2 x3

x1

AAAA

AAAA

A

PPPPPP

PPPPPP

PP

UUUUUUUU

UUUUUUUU

UUUUx2

}}}}

}}}}

}

AAAA

AAAA

A

PPPPPP

PPPPPP

PPx3

nnnnnn

nnnnnn

nn

}}}}

}}}}

}

AAAA

AAAA

A x4

iiiiiiii

iiiiiiii

iiii

nnnnnn

nnnnnn

nn

}}}}

}}}}

}

x1 x2 x3 x4

I Recursively defined

I `1 radix- switches to merge outputs of radix-subnetworks of dimension ` 1

I Merges ith output of each of the subnetworks

Example ( = 3, ` = 2)

0

1

\$\$III

IIII

II0

zzuuuuu

uuuu

0

0

\$\$III

IIII

II1

zzuuuuu

uuuu

0

\$\$III

IIII

II1

zzuuuuu

uuuu

1

0

++XXXXXXXXX

XXXXXXXXXX

XXX 0

++XXXXXXXXX

XXXXXXXXXX

XXX 1

0

++XXXXXXXXX

XXXXXXXXXX

XXX 1

ssffffffffff

ffffffffff

ff 0

1

qqcccccccccccccccccccc

cccccccccccccccccccc

ccc 0

1

1 1 1 0 0 0 0 0 1

I Recursively defined

I `1 radix- switches to merge outputs of radix-subnetworks of dimension ` 1

I Merges ith output of each of the subnetworks

Example ( = 3, ` = 2)

0

1

\$\$III

IIII

II0

zzuuuuu

uuuu

0

0

\$\$III

IIII

II1

zzuuuuu

uuuu

0

\$\$III

IIII

II1

zzuuuuu

uuuu

1

0

++XXXXXXXXX

XXXXXXXXXX

XXX 0

++XXXXXXXXX

XXXXXXXXXX

XXX 1

0

++XXXXXXXXX

XXXXXXXXXX

XXX 1

ssffffffffff

ffffffffff

ff 0

1

qqcccccccccccccccccccc

cccccccccccccccccccc

ccc 0

1

1 1 1 0 0 0 0 0 1

LemmaLet n = ` where 2. The `-dimensional radix- switchingnetwork can switch any r indices 1 i1 < < ir n into anydesired contiguous block of indices. For our purposes, wrappingthe block around the outside preserves contiguity.

Example ( = 3)

LemmaLet n = ` where 2. The `-dimensional radix- switchingnetwork can switch any r indices 1 i1 < < ir n into anydesired contiguous block of indices. For our purposes, wrappingthe block around the outside preserves contiguity.

Example ( = 3)

• Generalized Arbitrary Radix Switching Networks

I Let n =

pi=1

ni where

ni = ci

`i

ci {1, . . . , 1}`1 < `2 < < `p

I Radix-(ck + 1) switches to mergek1

i=1 ni subnetwork withfar right nodes of `k blocks

I Radix-ck switches to merge other nodes of `k blocks

• Generalized Arbitrary Radix Switching Networks

I Let n =

pi=1

ni where

ni = ci

`i

ci {1, . . . , 1}`1 < `2 < < `p

I Radix-(ck + 1) switches to mergek1

i=1 ni subnetwork withfar right nodes of `k blocks

I Radix-ck switches to merge other nodes of `k blocks

• Generalized Arbitrary Radix Switching Networks

Example ( = 3, n = 7 = 1 0 + 2 1)

0

0

1

\$\$III

IIII

II0

zzuuuuu

uuuu

0

**TTTTTTT

TTTTTTTT

T 1

zzuuuuu

uuuu

1

zzuuuuu

uuuu

0

++XXXXXXXXX

XXXXXXXXXX

XXX 0

++XXXXXXXXX

XXXXXXXXXX

XXX 0

++XXXXXXXXX

XXXXXXXXXX

XXX 1

ssffffffffff

ffffffffff

ff 1

ssffffffffff

ffffffffff

ff 1

ssffffffffff

ffffffffff

ff 0

1 1 1 0 0 0 0

TheoremSuppose 2. The generalized radix- switching networkdescribed above can switch any r indices 1 i1 < < ir n intothe contiguous block 1, 2, . . . , r . Furthermore, it has a depth ofdlog(n)e and a total of no more than dlog(n)edlog(n)e/switches of radix at most . The network attains this bound onlywhen n = dlog(n)e or n < .

• Generalized Arbitrary Radix Switching Networks

Example ( = 3, n = 7 = 1 0 + 2 1)

0

0

1

\$\$III

IIII

II0

zzuuuuu

uuuu

0

**TTTTTTT

TTTTTTTT

T 1

zzuuuuu

uuuu

1

zzuuuuu

uuuu

0

++XXXXXXXXX

XXXXXXXXXX

XXX 0

++XXXXXXXXX

XXXXXXXXXX

XXX 0

++XXXXXXXXX

XXXXXXXXXX

XXX 1

ssffffffffff

ffffffffff

ff 1

ssffffffffff

ffffffffff

ff 1

ssffffffffff

ffffffffff

ff 0

1 1 1 0 0 0 0

TheoremSuppose 2. The generalized radix- switching networkdescribed above can switch any r indices 1 i1 < < ir n intothe contiguous block 1, 2, . . . , r . Furthermore, it has a depth ofdlog(n)e and a total of no more than dlog(n)edlog(n)e/switches of radix at most . The network attains this bound onlywhen n = dlog(n)e or n < .

• Building Preconditioners

I Replace each switch by symbolic exchange matrix Sk tomerge columns (not mix)

I Embed each exchange matrix as principal minor of n nidentity matrix to create Sk

I Symbolic preconditioner is product: L =s

k=1 Sk

I Randomly choose values for symbols to create probabilisticpreconditioner: A = AL where L =

sk=1 Sk

• Building Preconditioners

Example ( = 3)

Sk =

1,1,k 1,2,k 1,3,k2,1,k 2,2,k 2,3,k3,1,k 3,2,k 3,3,k

Sk =

1. . .

1,1,k 1,2,k 1,3,k. . .

2,1,k 2,2,k 2,3,k. . .

3,1,k 3,2,k 3,3,k. . .

1

• Symbolic Toeplitz Matrix

T =

+1 . . . 21

1 . . . 22...

.... . .

...1 2 . . .

Using the lexicographic monomial ordering with1 2 2n1:

lt(det

(T[i1,...,im;j1,...,jm]

))= (1)bm/2c

mk=1

n+jm+1kik

• Symbolic Toeplitz Matrix

T =

+1 . . . 21

1 . . . 22...

.... . .

...1 2 . . .

Using the lexicographic monomial ordering with1 2 2n1:

lt(det

(T[i1,...,im;j1,...,jm]

))= (1)bm/2c

mk=1

n+jm+1kik

• Toeplitz Exchange Matrix

Let A be submatrix of A affected by T .

I A includes a subset of r of the r linearly independentcolumns of A where 0 r r .

I det((AT )[I ,J ]) =

K ={k1,...,km}1k1

• Toeplitz Exchange Matrix

Let A be submatrix of A affected by T .I A includes a subset of r of the r linearly independent

columns of A where 0 r r .

I det((AT )[I ,J ]) =

K ={k1,...,km}1k1

• Toeplitz Exchange Matrix

Let A be submatrix of A affected by T .I A includes a subset of r of the r linearly independent

columns of A where 0 r r .I det((AT )[I ,J ]) =

K ={k1,...,km}1k1

• Toeplitz Exchange Matrix

Let A be submatrix of A affected by T .I A includes a subset of r of the r linearly independent

columns of A where 0 r r .I det((AT )[I ,J ]) =

K ={k1,...,km}1k1

• Toeplitz Exchange Matrix

Let A be submatrix of A affected by T .I A includes a subset of r of the r linearly independent

columns of A where 0 r r .I det((AT )[I ,J ]) =

K ={k1,...,km}1k1

• Toeplitz Exchange Matrix

Let A be submatrix of A affected by T .I A includes a subset of r of the r linearly independent

columns of A where 0 r r .I det((AT )[I ,J ]) =

K ={k1,...,km}1k1

• Toeplitz Exchange Matrix

TheoremLet F be a field, let A Fnn have r linearly independent columns,let s be the number of switches in the generalized radix-switching network, and let S be a finite subset of F. Let N be thenumber of random numbers required in this network. If a1, . . . , aNare randomly cho