libflame revision_anchor
|
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) |
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 | ||
) |
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_lnbr().
{ /* 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; }