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

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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

  • Arbitrary Radix Switches

    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

  • Arbitrary Radix Switches

    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

  • Arbitrary Radix Switching Networks

    Definition (`-dimensional radix- switching network)

    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

  • Arbitrary Radix Switching Networks

    Definition (`-dimensional radix- switching network)

    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

  • Arbitrary Radix Switching Networks

    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)

  • Arbitrary Radix Switching Networks

    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

    Definition (Generalized radix- switching network)

    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

    Definition (Generalized radix- switching network)

    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