LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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◆ clarfb_gett()

 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

Purpose:
``` CLARFB_GETT applies a complex Householder block reflector H from the
left to a complex (K+M)-by-N  "triangular-pentagonal" 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.```
Parameters
 [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).```
Contributors:
``` November 2020, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```
Further Details:
```    (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 390 of file clarfb_gett.f.

392 IMPLICIT NONE
393*
394* -- LAPACK auxiliary routine --
395* -- LAPACK is a software package provided by Univ. of Tennessee, --
396* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
397*
398* .. Scalar Arguments ..
399 CHARACTER IDENT
400 INTEGER K, LDA, LDB, LDT, LDWORK, M, N
401* ..
402* .. Array Arguments ..
403 COMPLEX A( LDA, * ), B( LDB, * ), T( LDT, * ),
404 \$ WORK( LDWORK, * )
405* ..
406*
407* =====================================================================
408*
409* .. Parameters ..
410 COMPLEX CONE, CZERO
411 parameter( cone = ( 1.0e+0, 0.0e+0 ),
412 \$ czero = ( 0.0e+0, 0.0e+0 ) )
413* ..
414* .. Local Scalars ..
415 LOGICAL LNOTIDENT
416 INTEGER I, J
417* ..
418* .. EXTERNAL FUNCTIONS ..
419 LOGICAL LSAME
420 EXTERNAL lsame
421* ..
422* .. External Subroutines ..
423 EXTERNAL ccopy, cgemm, ctrmm
424* ..
425* .. Executable Statements ..
426*
427* Quick return if possible
428*
429 IF( m.LT.0 .OR. n.LE.0 .OR. k.EQ.0 .OR. k.GT.n )
430 \$ RETURN
431*
432 lnotident = .NOT.lsame( ident, 'I' )
433*
434* ------------------------------------------------------------------
435*
436* First Step. Computation of the Column Block 2:
437*
438* ( A2 ) := H * ( A2 )
439* ( B2 ) ( B2 )
440*
441* ------------------------------------------------------------------
442*
443 IF( n.GT.k ) THEN
444*
445* col2_(1) Compute W2: = A2. Therefore, copy A2 = A(1:K, K+1:N)
446* into W2=WORK(1:K, 1:N-K) column-by-column.
447*
448 DO j = 1, n-k
449 CALL ccopy( k, a( 1, k+j ), 1, work( 1, j ), 1 )
450 END DO
451
452 IF( lnotident ) THEN
453*
454* col2_(2) Compute W2: = (V1**H) * W2 = (A1**H) * W2,
455* V1 is not an identity matrix, but unit lower-triangular
456* V1 stored in A1 (diagonal ones are not stored).
457*
458*
459 CALL ctrmm( 'L', 'L', 'C', 'U', k, n-k, cone, a, lda,
460 \$ work, ldwork )
461 END IF
462*
463* col2_(3) Compute W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
464* V2 stored in B1.
465*
466 IF( m.GT.0 ) THEN
467 CALL cgemm( 'C', 'N', k, n-k, m, cone, b, ldb,
468 \$ b( 1, k+1 ), ldb, cone, work, ldwork )
469 END IF
470*
471* col2_(4) Compute W2: = T * W2,
472* T is upper-triangular.
473*
474 CALL ctrmm( 'L', 'U', 'N', 'N', k, n-k, cone, t, ldt,
475 \$ work, ldwork )
476*
477* col2_(5) Compute B2: = B2 - V2 * W2 = B2 - B1 * W2,
478* V2 stored in B1.
479*
480 IF( m.GT.0 ) THEN
481 CALL cgemm( 'N', 'N', m, n-k, k, -cone, b, ldb,
482 \$ work, ldwork, cone, b( 1, k+1 ), ldb )
483 END IF
484*
485 IF( lnotident ) THEN
486*
487* col2_(6) Compute W2: = V1 * W2 = A1 * W2,
488* V1 is not an identity matrix, but unit lower-triangular,
489* V1 stored in A1 (diagonal ones are not stored).
490*
491 CALL ctrmm( 'L', 'L', 'N', 'U', k, n-k, cone, a, lda,
492 \$ work, ldwork )
493 END IF
494*
495* col2_(7) Compute A2: = A2 - W2 =
496* = A(1:K, K+1:N-K) - WORK(1:K, 1:N-K),
497* column-by-column.
498*
499 DO j = 1, n-k
500 DO i = 1, k
501 a( i, k+j ) = a( i, k+j ) - work( i, j )
502 END DO
503 END DO
504*
505 END IF
506*
507* ------------------------------------------------------------------
508*
509* Second Step. Computation of the Column Block 1:
510*
511* ( A1 ) := H * ( A1 )
512* ( B1 ) ( 0 )
513*
514* ------------------------------------------------------------------
515*
516* col1_(1) Compute W1: = A1. Copy the upper-triangular
517* A1 = A(1:K, 1:K) into the upper-triangular
518* W1 = WORK(1:K, 1:K) column-by-column.
519*
520 DO j = 1, k
521 CALL ccopy( j, a( 1, j ), 1, work( 1, j ), 1 )
522 END DO
523*
524* Set the subdiagonal elements of W1 to zero column-by-column.
525*
526 DO j = 1, k - 1
527 DO i = j + 1, k
528 work( i, j ) = czero
529 END DO
530 END DO
531*
532 IF( lnotident ) THEN
533*
534* col1_(2) Compute W1: = (V1**H) * W1 = (A1**H) * W1,
535* V1 is not an identity matrix, but unit lower-triangular
536* V1 stored in A1 (diagonal ones are not stored),
537* W1 is upper-triangular with zeroes below the diagonal.
538*
539 CALL ctrmm( 'L', 'L', 'C', 'U', k, k, cone, a, lda,
540 \$ work, ldwork )
541 END IF
542*
543* col1_(3) Compute W1: = T * W1,
544* T is upper-triangular,
545* W1 is upper-triangular with zeroes below the diagonal.
546*
547 CALL ctrmm( 'L', 'U', 'N', 'N', k, k, cone, t, ldt,
548 \$ work, ldwork )
549*
550* col1_(4) Compute B1: = - V2 * W1 = - B1 * W1,
551* V2 = B1, W1 is upper-triangular with zeroes below the diagonal.
552*
553 IF( m.GT.0 ) THEN
554 CALL ctrmm( 'R', 'U', 'N', 'N', m, k, -cone, work, ldwork,
555 \$ b, ldb )
556 END IF
557*
558 IF( lnotident ) THEN
559*
560* col1_(5) Compute W1: = V1 * W1 = A1 * W1,
561* V1 is not an identity matrix, but unit lower-triangular
562* V1 stored in A1 (diagonal ones are not stored),
563* W1 is upper-triangular on input with zeroes below the diagonal,
564* and square on output.
565*
566 CALL ctrmm( 'L', 'L', 'N', 'U', k, k, cone, a, lda,
567 \$ work, ldwork )
568*
569* col1_(6) Compute A1: = A1 - W1 = A(1:K, 1:K) - WORK(1:K, 1:K)
570* column-by-column. A1 is upper-triangular on input.
571* If IDENT, A1 is square on output, and W1 is square,
572* if NOT IDENT, A1 is upper-triangular on output,
573* W1 is upper-triangular.
574*
575* col1_(6)_a Compute elements of A1 below the diagonal.
576*
577 DO j = 1, k - 1
578 DO i = j + 1, k
579 a( i, j ) = - work( i, j )
580 END DO
581 END DO
582*
583 END IF
584*
585* col1_(6)_b Compute elements of A1 on and above the diagonal.
586*
587 DO j = 1, k
588 DO i = 1, j
589 a( i, j ) = a( i, j ) - work( i, j )
590 END DO
591 END DO
592*
593 RETURN
594*
595* End of CLARFB_GETT
596*
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
subroutine cgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
CGEMM
Definition cgemm.f:188
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine ctrmm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb)
CTRMM
Definition ctrmm.f:177
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