 |
LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
|
| subroutine clarfb_gett | ( | character | ident, |
| integer | m, | ||
| integer | n, | ||
| integer | k, | ||
| complex, dimension( ldt, * ) | t, | ||
| integer | ldt, | ||
| complex, dimension( lda, * ) | a, | ||
| integer | lda, | ||
| complex, dimension( ldb, * ) | b, | ||
| integer | ldb, | ||
| complex, dimension( ldwork, * ) | work, | ||
| integer | ldwork ) |
CLARFB_GETT
Download CLARFB_GETT + dependencies [TGZ] [ZIP] [TXT]
!> !> CLARFB_GETT applies a complex Householder block reflector H from the !> left to a complex (K+M)-by-N matrix !> composed of two block matrices: an upper trapezoidal K-by-N matrix A !> stored in the array A, and a rectangular M-by-(N-K) matrix B, stored !> in the array B. The block reflector H is stored in a compact !> WY-representation, where the elementary reflectors are in the !> arrays A, B and T. See Further Details section. !>
| [in] | IDENT | !> IDENT is CHARACTER*1 !> If IDENT = not 'I', or not 'i', then V1 is unit !> lower-triangular and stored in the left K-by-K block of !> the input matrix A, !> If IDENT = 'I' or 'i', then V1 is an identity matrix and !> not stored. !> See Further Details section. !> |
| [in] | M | !> M is INTEGER !> The number of rows of the matrix B. !> M >= 0. !> |
| [in] | N | !> N is INTEGER !> The number of columns of the matrices A and B. !> N >= 0. !> |
| [in] | K | !> K is INTEGER !> The number or rows of the matrix A. !> K is also order of the matrix T, i.e. the number of !> elementary reflectors whose product defines the block !> reflector. 0 <= K <= N. !> |
| [in] | T | !> T is COMPLEX array, dimension (LDT,K) !> The upper-triangular K-by-K matrix T in the representation !> of the block reflector. !> |
| [in] | LDT | !> LDT is INTEGER !> The leading dimension of the array T. LDT >= K. !> |
| [in,out] | A | !> A is COMPLEX array, dimension (LDA,N) !> !> On entry: !> a) In the K-by-N upper-trapezoidal part A: input matrix A. !> b) In the columns below the diagonal: columns of V1 !> (ones are not stored on the diagonal). !> !> On exit: !> A is overwritten by rectangular K-by-N product H*A. !> !> See Further Details section. !> |
| [in] | LDA | !> LDB is INTEGER !> The leading dimension of the array A. LDA >= max(1,K). !> |
| [in,out] | B | !> B is COMPLEX array, dimension (LDB,N) !> !> On entry: !> a) In the M-by-(N-K) right block: input matrix B. !> b) In the M-by-N left block: columns of V2. !> !> On exit: !> B is overwritten by rectangular M-by-N product H*B. !> !> See Further Details section. !> |
| [in] | LDB | !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,M). !> |
| [out] | WORK | !> WORK is COMPLEX array, !> dimension (LDWORK,max(K,N-K)) !> |
| [in] | LDWORK | !> LDWORK is INTEGER !> The leading dimension of the array WORK. LDWORK>=max(1,K). !> !> |
!> !> November 2020, Igor Kozachenko, !> Computer Science Division, !> University of California, Berkeley !> !>
!> !> (1) Description of the Algebraic Operation. !> !> The matrix A is a K-by-N matrix composed of two column block !> matrices, A1, which is K-by-K, and A2, which is K-by-(N-K): !> A = ( A1, A2 ). !> The matrix B is an M-by-N matrix composed of two column block !> matrices, B1, which is M-by-K, and B2, which is M-by-(N-K): !> B = ( B1, B2 ). !> !> Perform the operation: !> !> ( A_out ) := H * ( A_in ) = ( I - V * T * V**H ) * ( A_in ) = !> ( B_out ) ( B_in ) ( B_in ) !> = ( I - ( V1 ) * T * ( V1**H, V2**H ) ) * ( A_in ) !> ( V2 ) ( B_in ) !> On input: !> !> a) ( A_in ) consists of two block columns: !> ( B_in ) !> !> ( A_in ) = (( A1_in ) ( A2_in )) = (( A1_in ) ( A2_in )) !> ( B_in ) (( B1_in ) ( B2_in )) (( 0 ) ( B2_in )), !> !> where the column blocks are: !> !> ( A1_in ) is a K-by-K upper-triangular matrix stored in the !> upper triangular part of the array A(1:K,1:K). !> ( B1_in ) is an M-by-K rectangular ZERO matrix and not stored. !> !> ( A2_in ) is a K-by-(N-K) rectangular matrix stored !> in the array A(1:K,K+1:N). !> ( B2_in ) is an M-by-(N-K) rectangular matrix stored !> in the array B(1:M,K+1:N). !> !> b) V = ( V1 ) !> ( V2 ) !> !> where: !> 1) if IDENT == 'I',V1 is a K-by-K identity matrix, not stored; !> 2) if IDENT != 'I',V1 is a K-by-K unit lower-triangular matrix, !> stored in the lower-triangular part of the array !> A(1:K,1:K) (ones are not stored), !> and V2 is an M-by-K rectangular stored the array B(1:M,1:K), !> (because on input B1_in is a rectangular zero !> matrix that is not stored and the space is !> used to store V2). !> !> c) T is a K-by-K upper-triangular matrix stored !> in the array T(1:K,1:K). !> !> On output: !> !> a) ( A_out ) consists of two block columns: !> ( B_out ) !> !> ( A_out ) = (( A1_out ) ( A2_out )) !> ( B_out ) (( B1_out ) ( B2_out )), !> !> where the column blocks are: !> !> ( A1_out ) is a K-by-K square matrix, or a K-by-K !> upper-triangular matrix, if V1 is an !> identity matrix. AiOut is stored in !> the array A(1:K,1:K). !> ( B1_out ) is an M-by-K rectangular matrix stored !> in the array B(1:M,K:N). !> !> ( A2_out ) is a K-by-(N-K) rectangular matrix stored !> in the array A(1:K,K+1:N). !> ( B2_out ) is an M-by-(N-K) rectangular matrix stored !> in the array B(1:M,K+1:N). !> !> !> The operation above can be represented as the same operation !> on each block column: !> !> ( A1_out ) := H * ( A1_in ) = ( I - V * T * V**H ) * ( A1_in ) !> ( B1_out ) ( 0 ) ( 0 ) !> !> ( A2_out ) := H * ( A2_in ) = ( I - V * T * V**H ) * ( A2_in ) !> ( B2_out ) ( B2_in ) ( B2_in ) !> !> If IDENT != 'I': !> !> The computation for column block 1: !> !> A1_out: = A1_in - V1*T*(V1**H)*A1_in !> !> B1_out: = - V2*T*(V1**H)*A1_in !> !> The computation for column block 2, which exists if N > K: !> !> A2_out: = A2_in - V1*T*( (V1**H)*A2_in + (V2**H)*B2_in ) !> !> B2_out: = B2_in - V2*T*( (V1**H)*A2_in + (V2**H)*B2_in ) !> !> If IDENT == 'I': !> !> The operation for column block 1: !> !> A1_out: = A1_in - V1*T*A1_in !> !> B1_out: = - V2*T*A1_in !> !> The computation for column block 2, which exists if N > K: !> !> A2_out: = A2_in - T*( A2_in + (V2**H)*B2_in ) !> !> B2_out: = B2_in - V2*T*( A2_in + (V2**H)*B2_in ) !> !> (2) Description of the Algorithmic Computation. !> !> In the first step, we compute column block 2, i.e. A2 and B2. !> Here, we need to use the K-by-(N-K) rectangular workspace !> matrix W2 that is of the same size as the matrix A2. !> W2 is stored in the array WORK(1:K,1:(N-K)). !> !> In the second step, we compute column block 1, i.e. A1 and B1. !> Here, we need to use the K-by-K square workspace matrix W1 !> that is of the same size as the as the matrix A1. !> W1 is stored in the array WORK(1:K,1:K). !> !> NOTE: Hence, in this routine, we need the workspace array WORK !> only of size WORK(1:K,1:max(K,N-K)) so it can hold both W2 from !> the first step and W1 from the second step. !> !> Case (A), when V1 is unit lower-triangular, i.e. IDENT != 'I', !> more computations than in the Case (B). !> !> if( IDENT != 'I' ) then !> if ( N > K ) then !> (First Step - column block 2) !> col2_(1) W2: = A2 !> col2_(2) W2: = (V1**H) * W2 = (unit_lower_tr_of_(A1)**H) * W2 !> col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2 !> col2_(4) W2: = T * W2 !> col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2 !> col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2 !> col2_(7) A2: = A2 - W2 !> else !> (Second Step - column block 1) !> col1_(1) W1: = A1 !> col1_(2) W1: = (V1**H) * W1 = (unit_lower_tr_of_(A1)**H) * W1 !> col1_(3) W1: = T * W1 !> col1_(4) B1: = - V2 * W1 = - B1 * W1 !> col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1 !> col1_(6) square A1: = A1 - W1 !> end if !> end if !> !> Case (B), when V1 is an identity matrix, i.e. IDENT == 'I', !> less computations than in the Case (A) !> !> if( IDENT == 'I' ) then !> if ( N > K ) then !> (First Step - column block 2) !> col2_(1) W2: = A2 !> col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2 !> col2_(4) W2: = T * W2 !> col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2 !> col2_(7) A2: = A2 - W2 !> else !> (Second Step - column block 1) !> col1_(1) W1: = A1 !> col1_(3) W1: = T * W1 !> col1_(4) B1: = - V2 * W1 = - B1 * W1 !> col1_(6) upper-triangular_of_(A1): = A1 - W1 !> end if !> end if !> !> Combine these cases (A) and (B) together, this is the resulting !> algorithm: !> !> if ( N > K ) then !> !> (First Step - column block 2) !> !> col2_(1) W2: = A2 !> if( IDENT != 'I' ) then !> col2_(2) W2: = (V1**H) * W2 !> = (unit_lower_tr_of_(A1)**H) * W2 !> end if !> col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2] !> col2_(4) W2: = T * W2 !> col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2 !> if( IDENT != 'I' ) then !> col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2 !> end if !> col2_(7) A2: = A2 - W2 !> !> else !> !> (Second Step - column block 1) !> !> col1_(1) W1: = A1 !> if( IDENT != 'I' ) then !> col1_(2) W1: = (V1**H) * W1 !> = (unit_lower_tr_of_(A1)**H) * W1 !> end if !> col1_(3) W1: = T * W1 !> col1_(4) B1: = - V2 * W1 = - B1 * W1 !> if( IDENT != 'I' ) then !> col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1 !> col1_(6_a) below_diag_of_(A1): = - below_diag_of_(W1) !> end if !> col1_(6_b) up_tr_of_(A1): = up_tr_of_(A1) - up_tr_of_(W1) !> !> end if !> !>
Definition at line 388 of file clarfb_gett.f.
1.11.0 
