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Functions
FLA_Apply_Q_UT_lhfc_blk_var1.c File Reference

(r)

Functions

FLA_Error FLA_Apply_Q_UT_lhfc_blk_var1 (FLA_Obj A, FLA_Obj T, FLA_Obj W, FLA_Obj B, fla_apqut_t *cntl)

Function Documentation

FLA_Error FLA_Apply_Q_UT_lhfc_blk_var1 ( FLA_Obj  A,
FLA_Obj  T,
FLA_Obj  W,
FLA_Obj  B,
fla_apqut_t cntl 
)

References FLA_Axpyt_internal(), FLA_Cont_with_1x3_to_1x2(), FLA_Cont_with_3x1_to_2x1(), FLA_Cont_with_3x3_to_2x2(), FLA_Copyt_internal(), FLA_Gemm_internal(), FLA_MINUS_ONE, FLA_Obj_length(), FLA_Obj_min_dim(), FLA_Obj_width(), FLA_ONE, FLA_Part_1x2(), FLA_Part_2x1(), FLA_Part_2x2(), FLA_Repart_1x2_to_1x3(), FLA_Repart_2x1_to_3x1(), FLA_Repart_2x2_to_3x3(), FLA_Trmm_internal(), and FLA_Trsm_internal().

Referenced by FLA_Apply_Q_UT_lhfc().

{
/*
  Apply the conjugate-transpose of a unitary matrix Q to a matrix B from the
  left,

    B := Q' B

  where Q is the forward product of Householder transformations:

    Q  =  H(0) H(1) ... H(k-1)

  where H(i) corresponds to the Householder vector stored below the diagonal
  in the ith column of A. Thus, the operation becomes:

    B :=  Q' B
       =  ( H(0) H(1) ... H(k-1) )' B
       =  H(k-1)' ... H(1)' H(0)' B

  From this, we can see that we must move through A from top-left to bottom-
  right, since the Householder vector for H(0) was stored in the first column
  of A. We intend to apply blocks of reflectors at a time, where a block
  reflector H of b consecutive Householder transforms may be expressed as:

    H  =  ( H(i) H(i+1) ... H(i+b-1) )'
       =  ( I - U inv(T) U' )'

  where:
    - U is the strictly lower trapezoidal (with implicit unit diagonal) matrix
      of Householder vectors, stored below the diagonal of A in columns i
      through i+b-1, corresponding to H(i) through H(i+b-1).
    - T is the upper triangular block Householder matrix corresponding to
      Householder vectors i through i+b-1.

  Consider applying H to B as an intermediate step towards applying all of Q':

    B  :=  H B
        =  ( I - U inv(T) U' )' B
        =  ( I - U inv(T)' U' ) B
        =  B - U inv(T)' U' B

  We must move from top-left to bottom-right. So, we partition:

    U -> / U11 \  B -> / B1 \  T -> ( T1 T2 )
         \ U21 /       \ B2 /

  where:
    - U11 is stored in the strictly lower triangle of A11 with implicit unit
      diagonal.
    - U21 is stored in A21.
    - T1 is an upper triangular block of row-panel matrix T.

  Substituting repartitioned U, B, and T, we have:

    / B1 \  :=   / B1 \ - / U11 \ inv(T1)' / U11 \' / B1 \
    \ B2 /       \ B2 /   \ U21 /          \ U21 /  \ B2 /
             =   / B1 \ - / U11 \ inv(T1)' ( U11' U21' ) / B1 \
                 \ B2 /   \ U21 /                        \ B2 /
             =   / B1 \ - / U11 \ inv(T1)' ( U11' B1 + U21' B2 )
                 \ B2 /   \ U21 /

  Thus, B1 is updated as:

      B1    :=     B1   -   U11 inv(T1)' ( U11' B1 + U21' B2 )

  And B2 is updated as:

      B2    :=     B2   -   U21 inv(T1)' ( U11' B1 + U21' B2 )

  Note that:

    inv(T1)' ( U11' B1 + U21' B2 )

  is common to both updates, and thus may be computed and stored in
  workspace, and then re-used.

  -FGVZ
*/
  FLA_Obj ATL,   ATR,      A00, A01, A02, 
          ABL,   ABR,      A10, A11, A12,
                           A20, A21, A22;

  FLA_Obj TL,    TR,       T0,  T1,  T2;

  FLA_Obj T1T,
          T2B;

  FLA_Obj WTL,  WTR,
          WBL,  WBR;

  FLA_Obj BT,              B0,
          BB,              B1,
                           B2;

  dim_t   b_alg, b;

  // Query the algorithmic blocksize by inspecting the length of T.
  b_alg = FLA_Obj_length( T );

  FLA_Part_2x2( A,    &ATL, &ATR,
                      &ABL, &ABR,     0, 0, FLA_TL );

  FLA_Part_1x2( T,    &TL,  &TR,      0, FLA_LEFT );

  FLA_Part_2x1( B,    &BT, 
                      &BB,            0, FLA_TOP );

  while ( FLA_Obj_min_dim( ABR ) > 0 ){

    b = min( b_alg, FLA_Obj_min_dim( ABR ) );

    FLA_Repart_2x2_to_3x3( ATL, /**/ ATR,       &A00, /**/ &A01, &A02,
                        /* ************* */   /* ******************** */
                                                &A10, /**/ &A11, &A12,
                           ABL, /**/ ABR,       &A20, /**/ &A21, &A22,
                           b, b, FLA_BR );

    FLA_Repart_1x2_to_1x3( TL,  /**/ TR,        &T0, /**/ &T1, &T2,
                           b, FLA_RIGHT );

    FLA_Repart_2x1_to_3x1( BT,                &B0, 
                        /* ** */            /* ** */
                                              &B1, 
                           BB,                &B2,        b, FLA_BOTTOM );

    /*------------------------------------------------------------*/

    FLA_Part_2x1( T1,    &T1T, 
                         &T2B,     b, FLA_TOP );

    FLA_Part_2x2( W,     &WTL, &WTR,
                         &WBL, &WBR,     b, FLA_Obj_width( B1 ), FLA_TL );

    // WTL = B1;

    FLA_Copyt_internal( FLA_NO_TRANSPOSE, B1, WTL,
                        FLA_Cntl_sub_copyt( cntl ) );

    // U11 = trilu( A11 );
    // U21 = A21;
    //
    // WTL = inv( triu(T1T) )' * ( U11' * B1 + U21' * B2 );

    FLA_Trmm_internal( FLA_LEFT, FLA_LOWER_TRIANGULAR,
                       FLA_CONJ_TRANSPOSE, FLA_UNIT_DIAG,
                       FLA_ONE, A11, WTL,
                       FLA_Cntl_sub_trmm1( cntl ) );

    FLA_Gemm_internal( FLA_CONJ_TRANSPOSE, FLA_NO_TRANSPOSE, 
                       FLA_ONE, A21, B2, FLA_ONE, WTL,
                       FLA_Cntl_sub_gemm1( cntl ) );

    FLA_Trsm_internal( FLA_LEFT, FLA_UPPER_TRIANGULAR,
                       FLA_CONJ_TRANSPOSE, FLA_NONUNIT_DIAG,
                       FLA_ONE, T1T, WTL,
                       FLA_Cntl_sub_trsm( cntl ) );

    // B2 = B2 - U21 * WTL;
    // B1 = B1 - U11 * WTL;

    FLA_Gemm_internal( FLA_NO_TRANSPOSE, FLA_NO_TRANSPOSE,
                       FLA_MINUS_ONE, A21, WTL, FLA_ONE, B2,
                       FLA_Cntl_sub_gemm2( cntl ) );

    FLA_Trmm_internal( FLA_LEFT, FLA_LOWER_TRIANGULAR,
                       FLA_NO_TRANSPOSE, FLA_UNIT_DIAG,
                       FLA_MINUS_ONE, A11, WTL,
                       FLA_Cntl_sub_trmm2( cntl ) );

    FLA_Axpyt_internal( FLA_NO_TRANSPOSE, FLA_ONE, WTL, B1,
                        FLA_Cntl_sub_axpyt( cntl ) );

    /*------------------------------------------------------------*/

    FLA_Cont_with_3x3_to_2x2( &ATL, /**/ &ATR,       A00, A01, /**/ A02,
                                                     A10, A11, /**/ A12,
                            /* ************** */  /* ****************** */
                              &ABL, /**/ &ABR,       A20, A21, /**/ A22,
                              FLA_TL );

    FLA_Cont_with_1x3_to_1x2( &TL,  /**/ &TR,        T0, T1, /**/ T2,
                              FLA_LEFT );

    FLA_Cont_with_3x1_to_2x1( &BT,                B0, 
                                                  B1, 
                            /* ** */           /* ** */
                              &BB,                B2,     FLA_TOP );
  }

  return FLA_SUCCESS;
}