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Functions | |
| FLA_Error | FLA_Apply_Q_UT_lhfr_blk_var1 (FLA_Obj A, FLA_Obj T, FLA_Obj W, FLA_Obj B, fla_apqut_t *cntl) |
| FLA_Error | FLA_Apply_Q_UT_lhfr_blk_var2 (FLA_Obj A, FLA_Obj T, FLA_Obj W, FLA_Obj B, fla_apqut_t *cntl) |
| FLA_Error | FLA_Apply_Q_UT_lhfr_blk_var3 (FLA_Obj A, FLA_Obj T, FLA_Obj W, FLA_Obj B, fla_apqut_t *cntl) |
| FLA_Error FLA_Apply_Q_UT_lhfr_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_lhfr().
{
/*
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 above the diagonal
in the ith row 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 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;
}
| FLA_Error FLA_Apply_Q_UT_lhfr_blk_var2 | ( | FLA_Obj | A, |
| FLA_Obj | T, | ||
| FLA_Obj | W, | ||
| FLA_Obj | B, | ||
| fla_apqut_t * | cntl | ||
| ) |
References FLA_Apply_Q_UT_internal(), FLA_Cont_with_1x3_to_1x2(), FLA_Determine_blocksize(), FLA_Obj_width(), FLA_Part_1x2(), and FLA_Repart_1x2_to_1x3().
Referenced by FLA_Apply_Q_UT_lhfr().
{
FLA_Obj BL, BR, B0, B1, B2;
FLA_Obj WL, WR, W0, W1, W2;
dim_t b;
FLA_Part_1x2( B, &BL, &BR, 0, FLA_LEFT );
FLA_Part_1x2( W, &WL, &WR, 0, FLA_LEFT );
while ( FLA_Obj_width( BL ) < FLA_Obj_width( B ) ){
b = FLA_Determine_blocksize( BR, FLA_RIGHT, FLA_Cntl_blocksize( cntl ) );
FLA_Repart_1x2_to_1x3( BL, /**/ BR, &B0, /**/ &B1, &B2,
b, FLA_RIGHT );
FLA_Repart_1x2_to_1x3( WL, /**/ WR, &W0, /**/ &W1, &W2,
b, FLA_RIGHT );
/*------------------------------------------------------------*/
// B1 = Q' * B1;
FLA_Apply_Q_UT_internal( FLA_LEFT, FLA_CONJ_TRANSPOSE, FLA_FORWARD, FLA_ROWWISE,
A, T, W1, B1,
FLA_Cntl_sub_apqut( cntl ) );
/*------------------------------------------------------------*/
FLA_Cont_with_1x3_to_1x2( &BL, /**/ &BR, B0, B1, /**/ B2,
FLA_LEFT );
FLA_Cont_with_1x3_to_1x2( &WL, /**/ &WR, W0, W1, /**/ W2,
FLA_LEFT );
}
return FLA_SUCCESS;
}
| FLA_Error FLA_Apply_Q_UT_lhfr_blk_var3 | ( | FLA_Obj | A, |
| FLA_Obj | T, | ||
| FLA_Obj | W, | ||
| FLA_Obj | B, | ||
| fla_apqut_t * | cntl | ||
| ) |
References FLA_Axpyt_internal(), FLA_Cont_with_3x1_to_2x1(), FLA_Cont_with_3x3_to_2x2(), FLA_Copyt_internal(), FLA_Determine_blocksize(), FLA_Gemm_internal(), FLA_MINUS_ONE, FLA_Obj_min_dim(), FLA_Obj_width(), FLA_ONE, FLA_Part_2x1(), FLA_Part_2x2(), FLA_Repart_2x1_to_3x1(), FLA_Repart_2x2_to_3x3(), FLA_Trmm_internal(), and FLA_Trsm_internal().
Referenced by FLA_Apply_Q_UT_lhfr().
{
FLA_Obj ATL, ATR, A00, A01, A02,
ABL, ABR, A10, A11, A12,
A20, A21, A22;
FLA_Obj TWTL, TWTR, TW00, TW01, TW02,
TWBL, TWBR, TW10, T11, W12,
TW20, TW21, TW22;
FLA_Obj WTL, WTR,
WBL, WBR;
FLA_Obj BT, B0,
BB, B1,
B2;
dim_t b;
FLA_Part_2x2( A, &ATL, &ATR,
&ABL, &ABR, 0, 0, FLA_TL );
FLA_Part_2x2( TW, &TWTL, &TWTR,
&TWBL, &TWBR, 0, 0, FLA_TL );
FLA_Part_2x1( B, &BT,
&BB, 0, FLA_TOP );
while ( FLA_Obj_min_dim( ABR ) > 0 ){
b = FLA_Determine_blocksize( ABR, FLA_BR, FLA_Cntl_blocksize( cntl ) );
FLA_Repart_2x2_to_3x3( ATL, /**/ ATR, &A00, /**/ &A01, &A02,
/* ************* */ /* ******************** */
&A10, /**/ &A11, &A12,
ABL, /**/ ABR, &A20, /**/ &A21, &A22,
b, b, FLA_BR );
FLA_Repart_2x2_to_3x3( TWTL, /**/ TWTR, &TW00, /**/ &TW01, &TW02,
/* *************** */ /* *********************** */
&TW10, /**/ &T11, &W12,
TWBL, /**/ TWBR, &TW20, /**/ &TW21, &TW22,
b, b, FLA_BR );
FLA_Repart_2x1_to_3x1( BT, &B0,
/* ** */ /* ** */
&B1,
BB, &B2, b, FLA_BOTTOM );
/*------------------------------------------------------------*/
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(T11) )' * ( 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, T11, 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_3x3_to_2x2( &TWTL, /**/ &TWTR, TW00, TW01, /**/ TW02,
TW10, T11, /**/ W12,
/* **************** */ /* ********************* */
&TWBL, /**/ &TWBR, TW20, TW21, /**/ TW22,
FLA_TL );
FLA_Cont_with_3x1_to_2x1( &BT, B0,
B1,
/* ** */ /* ** */
&BB, B2, FLA_TOP );
}
return FLA_SUCCESS;
}
1.7.6.1