FLA_Apply_Q_UT_lnbr_blk_var1.c File Reference

(r)

Functions
FLA_Error	FLA_Apply_Q_UT_lnbr_blk_var1 (FLA_Obj A, FLA_Obj T, FLA_Obj W, FLA_Obj B, fla_apqut_t *cntl)

Function Documentation

FLA_Apply_Q_UT_lnbr_blk_var1()

FLA_Error FLA_Apply_Q_UT_lnbr_blk_var1	(	FLA_Obj	A,
		FLA_Obj	T,
		FLA_Obj	W,
		FLA_Obj	B,
		fla_apqut_t *	cntl
	)

{
/*
  Apply a unitary matrix Q to a matrix B from the left,
 
    B := Q B
 
  where Q is the backward product of Householder transformations:
 
    Q  =  H(k-1) ... H(1) H(0)
 
  where H(i) corresponds to the Householder vector stored above the diagonal
  in the ith row of A. Thus, the operation becomes:
 
    B :=  Q B
       =  H(k-1) ... H(1) H(0) B
       =  H(k-1)' ... H(1)' H(0)' B
       =  ( H(0) H(1) ... H(k-1) )' 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 row
  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^T is the strictly upper trapezoidal (with implicit unit diagonal) matrix
      of Householder vectors, stored above the diagonal of A in rows 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^T -> ( U11 U12 )  B -> / B1 \  T -> ( T1 T2 )
                             \ B2 /
 
  where:
    - U11 is stored in the strictly upper triangle of A11 with implicit unit
      diagonal.
    - U12 is stored in A12.
    - T1 is an upper triangular block of row-panel matrix T.
 
  Substituting repartitioned U, B, and T, we have:
 
    / B1 \  :=   / B1 \ - ( U11 U12 )^T inv(T1)' conj( U11 U12 ) / B1 \
    \ B2 /       \ B2 /                                          \ B2 /
             =   / B1 \ - / U11^T \ inv(T1)' conj( U11 U12 ) / B1 \
                 \ B2 /   \ U12^T /                          \ B2 /
             =   / B1 \ - / U11^T \ inv(T1)' ( conj(U11) B1 + conj(U12) B2 )
                 \ B2 /   \ U12^T /
 
  Thus, B1 is updated as:
 
      B1    :=     B1   -   U11^T inv(T1)' ( conj(U11) B1 + conj(U12) B2 )
 
  And B2 is updated as:
 
      B2    :=     B2   -   U12^T inv(T1)' ( conj(U11) B1 + conj(U12) B2 )
 
  Note that:
 
    inv(T1)' ( conj(U11) B1 + conj(U12) 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 = triuu( A11 );
    // U12 = A12;
    //
    // WTL = inv( triu(T1T) )' * ( conj(U11) * B1 + conj(U12) * B2 );
 
    FLA_Trmm_internal( FLA_LEFT, FLA_UPPER_TRIANGULAR,
                       FLA_CONJ_NO_TRANSPOSE, FLA_UNIT_DIAG,
                       FLA_ONE, A11, WTL,
                       FLA_Cntl_sub_trmm1( cntl ) );
 
    FLA_Gemm_internal( FLA_CONJ_NO_TRANSPOSE, FLA_NO_TRANSPOSE, 
                       FLA_ONE, A12, 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 - U12^T * WTL;
    // B1 = B1 - U11^T * WTL;
 
    FLA_Gemm_internal( FLA_TRANSPOSE, FLA_NO_TRANSPOSE,
                       FLA_MINUS_ONE, A12, WTL, FLA_ONE, B2,
                       FLA_Cntl_sub_gemm2( cntl ) );
 
    FLA_Trmm_internal( FLA_LEFT, FLA_UPPER_TRIANGULAR,
                       FLA_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;
}

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(), FLA_Trsm_internal(), and i.

Referenced by FLA_Apply_Q_UT_lnbr().