.TH "unhr_col" 3 "Sat Dec 9 2023 21:42:18" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME unhr_col \- {un,or}hr_col: Householder reconstruction .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBcunhr_col\fP (m, n, nb, a, lda, t, ldt, d, info)" .br .RI "\fBCUNHR_COL\fP " .ti -1c .RI "subroutine \fBdorhr_col\fP (m, n, nb, a, lda, t, ldt, d, info)" .br .RI "\fBDORHR_COL\fP " .ti -1c .RI "subroutine \fBsorhr_col\fP (m, n, nb, a, lda, t, ldt, d, info)" .br .RI "\fBSORHR_COL\fP " .ti -1c .RI "subroutine \fBzunhr_col\fP (m, n, nb, a, lda, t, ldt, d, info)" .br .RI "\fBZUNHR_COL\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine cunhr_col (integer m, integer n, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( * ) d, integer info)" .PP \fBCUNHR_COL\fP .PP \fBPurpose:\fP .RS 4 .PP .nf CUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns as input, stored in A, and performs Householder Reconstruction (HR), i\&.e\&. reconstructs Householder vectors V(i) implicitly representing another M-by-N matrix Q_out, with the property that Q_in = Q_out*S, where S is an N-by-N diagonal matrix with diagonal entries equal to +1 or -1\&. The Householder vectors (columns V(i) of V) are stored in A on output, and the diagonal entries of S are stored in D\&. Block reflectors are also returned in T (same output format as CGEQRT)\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. M >= N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The column block size to be used in the reconstruction of Householder column vector blocks in the array A and corresponding block reflectors in the array T\&. NB >= 1\&. (Note that if NB > N, then N is used instead of NB as the column block size\&.) .fi .PP .br \fIA\fP .PP .nf A is COMPLEX array, dimension (LDA,N) On entry: The array A contains an M-by-N orthonormal matrix Q_in, i\&.e the columns of A are orthogonal unit vectors\&. On exit: The elements below the diagonal of A represent the unit lower-trapezoidal matrix V of Householder column vectors V(i)\&. The unit diagonal entries of V are not stored (same format as the output below the diagonal in A from CGEQRT)\&. The matrix T and the matrix V stored on output in A implicitly define Q_out\&. The elements above the diagonal contain the factor U of the 'modified' LU-decomposition: Q_in - ( S ) = V * U ( 0 ) where 0 is a (M-N)-by-(M-N) zero matrix\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fIT\fP .PP .nf T is COMPLEX array, dimension (LDT, N) Let NOCB = Number_of_output_col_blocks = CEIL(N/NB) On exit, T(1:NB, 1:N) contains NOCB upper-triangular block reflectors used to define Q_out stored in compact form as a sequence of upper-triangular NB-by-NB column blocks (same format as the output T in CGEQRT)\&. The matrix T and the matrix V stored on output in A implicitly define Q_out\&. NOTE: The lower triangles below the upper-triangular blocks will be filled with zeros\&. See Further Details\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,min(NB,N))\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX array, dimension min(M,N)\&. The elements can be only plus or minus one\&. D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where 1 <= i <= min(M,N), and Q_in_i is Q_in after performing i-1 steps of “modified” Gaussian elimination\&. See Further Details\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf The computed M-by-M unitary factor Q_out is defined implicitly as a product of unitary matrices Q_out(i)\&. Each Q_out(i) is stored in the compact WY-representation format in the corresponding blocks of matrices V (stored in A) and T\&. The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N matrix A contains the column vectors V(i) in NB-size column blocks VB(j)\&. For example, VB(1) contains the columns V(1), V(2), \&.\&.\&. V(NB)\&. NOTE: The unit entries on the diagonal of Y are not stored in A\&. The number of column blocks is NOCB = Number_of_output_col_blocks = CEIL(N/NB) where each block is of order NB except for the last block, which is of order LAST_NB = N - (NOCB-1)*NB\&. For example, if M=6, N=5 and NB=2, the matrix V is V = ( VB(1), VB(2), VB(3) ) = = ( 1 ) ( v21 1 ) ( v31 v32 1 ) ( v41 v42 v43 1 ) ( v51 v52 v53 v54 1 ) ( v61 v62 v63 v54 v65 ) For each of the column blocks VB(i), an upper-triangular block reflector TB(i) is computed\&. These blocks are stored as a sequence of upper-triangular column blocks in the NB-by-N matrix T\&. The size of each TB(i) block is NB-by-NB, except for the last block, whose size is LAST_NB-by-LAST_NB\&. For example, if M=6, N=5 and NB=2, the matrix T is T = ( TB(1), TB(2), TB(3) ) = = ( t11 t12 t13 t14 t15 ) ( t22 t24 ) The M-by-M factor Q_out is given as a product of NOCB unitary M-by-M matrices Q_out(i)\&. Q_out = Q_out(1) * Q_out(2) * \&.\&.\&. * Q_out(NOCB), where each matrix Q_out(i) is given by the WY-representation using corresponding blocks from the matrices V and T: Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T, where I is the identity matrix\&. Here is the formula with matrix dimensions: Q(i){M-by-M} = I{M-by-M} - VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M}, where INB = NB, except for the last block NOCB for which INB=LAST_NB\&. ===== NOTE: ===== If Q_in is the result of doing a QR factorization B = Q_in * R_in, then: B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out\&. So if one wants to interpret Q_out as the result of the QR factorization of B, then the corresponding R_out should be equal to R_out = S * R_in, i\&.e\&. some rows of R_in should be multiplied by -1\&. For the details of the algorithm, see [1]\&. [1] 'Reconstructing Householder vectors from tall-skinny QR', G\&. Ballard, J\&. Demmel, L\&. Grigori, M\&. Jacquelin, H\&.D\&. Nguyen, E\&. Solomonik, J\&. Parallel Distrib\&. Comput\&., vol\&. 85, pp\&. 3-31, 2015\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley .fi .PP .RE .PP .SS "subroutine dorhr_col (integer m, integer n, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) d, integer info)" .PP \fBDORHR_COL\fP .PP \fBPurpose:\fP .RS 4 .PP .nf DORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns as input, stored in A, and performs Householder Reconstruction (HR), i\&.e\&. reconstructs Householder vectors V(i) implicitly representing another M-by-N matrix Q_out, with the property that Q_in = Q_out*S, where S is an N-by-N diagonal matrix with diagonal entries equal to +1 or -1\&. The Householder vectors (columns V(i) of V) are stored in A on output, and the diagonal entries of S are stored in D\&. Block reflectors are also returned in T (same output format as DGEQRT)\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. M >= N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The column block size to be used in the reconstruction of Householder column vector blocks in the array A and corresponding block reflectors in the array T\&. NB >= 1\&. (Note that if NB > N, then N is used instead of NB as the column block size\&.) .fi .PP .br \fIA\fP .PP .nf A is DOUBLE PRECISION array, dimension (LDA,N) On entry: The array A contains an M-by-N orthonormal matrix Q_in, i\&.e the columns of A are orthogonal unit vectors\&. On exit: The elements below the diagonal of A represent the unit lower-trapezoidal matrix V of Householder column vectors V(i)\&. The unit diagonal entries of V are not stored (same format as the output below the diagonal in A from DGEQRT)\&. The matrix T and the matrix V stored on output in A implicitly define Q_out\&. The elements above the diagonal contain the factor U of the 'modified' LU-decomposition: Q_in - ( S ) = V * U ( 0 ) where 0 is a (M-N)-by-(M-N) zero matrix\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fIT\fP .PP .nf T is DOUBLE PRECISION array, dimension (LDT, N) Let NOCB = Number_of_output_col_blocks = CEIL(N/NB) On exit, T(1:NB, 1:N) contains NOCB upper-triangular block reflectors used to define Q_out stored in compact form as a sequence of upper-triangular NB-by-NB column blocks (same format as the output T in DGEQRT)\&. The matrix T and the matrix V stored on output in A implicitly define Q_out\&. NOTE: The lower triangles below the upper-triangular blocks will be filled with zeros\&. See Further Details\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,min(NB,N))\&. .fi .PP .br \fID\fP .PP .nf D is DOUBLE PRECISION array, dimension min(M,N)\&. The elements can be only plus or minus one\&. D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where 1 <= i <= min(M,N), and Q_in_i is Q_in after performing i-1 steps of “modified” Gaussian elimination\&. See Further Details\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf The computed M-by-M orthogonal factor Q_out is defined implicitly as a product of orthogonal matrices Q_out(i)\&. Each Q_out(i) is stored in the compact WY-representation format in the corresponding blocks of matrices V (stored in A) and T\&. The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N matrix A contains the column vectors V(i) in NB-size column blocks VB(j)\&. For example, VB(1) contains the columns V(1), V(2), \&.\&.\&. V(NB)\&. NOTE: The unit entries on the diagonal of Y are not stored in A\&. The number of column blocks is NOCB = Number_of_output_col_blocks = CEIL(N/NB) where each block is of order NB except for the last block, which is of order LAST_NB = N - (NOCB-1)*NB\&. For example, if M=6, N=5 and NB=2, the matrix V is V = ( VB(1), VB(2), VB(3) ) = = ( 1 ) ( v21 1 ) ( v31 v32 1 ) ( v41 v42 v43 1 ) ( v51 v52 v53 v54 1 ) ( v61 v62 v63 v54 v65 ) For each of the column blocks VB(i), an upper-triangular block reflector TB(i) is computed\&. These blocks are stored as a sequence of upper-triangular column blocks in the NB-by-N matrix T\&. The size of each TB(i) block is NB-by-NB, except for the last block, whose size is LAST_NB-by-LAST_NB\&. For example, if M=6, N=5 and NB=2, the matrix T is T = ( TB(1), TB(2), TB(3) ) = = ( t11 t12 t13 t14 t15 ) ( t22 t24 ) The M-by-M factor Q_out is given as a product of NOCB orthogonal M-by-M matrices Q_out(i)\&. Q_out = Q_out(1) * Q_out(2) * \&.\&.\&. * Q_out(NOCB), where each matrix Q_out(i) is given by the WY-representation using corresponding blocks from the matrices V and T: Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T, where I is the identity matrix\&. Here is the formula with matrix dimensions: Q(i){M-by-M} = I{M-by-M} - VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M}, where INB = NB, except for the last block NOCB for which INB=LAST_NB\&. ===== NOTE: ===== If Q_in is the result of doing a QR factorization B = Q_in * R_in, then: B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out\&. So if one wants to interpret Q_out as the result of the QR factorization of B, then the corresponding R_out should be equal to R_out = S * R_in, i\&.e\&. some rows of R_in should be multiplied by -1\&. For the details of the algorithm, see [1]\&. [1] 'Reconstructing Householder vectors from tall-skinny QR', G\&. Ballard, J\&. Demmel, L\&. Grigori, M\&. Jacquelin, H\&.D\&. Nguyen, E\&. Solomonik, J\&. Parallel Distrib\&. Comput\&., vol\&. 85, pp\&. 3-31, 2015\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley .fi .PP .RE .PP .SS "subroutine sorhr_col (integer m, integer n, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) d, integer info)" .PP \fBSORHR_COL\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns as input, stored in A, and performs Householder Reconstruction (HR), i\&.e\&. reconstructs Householder vectors V(i) implicitly representing another M-by-N matrix Q_out, with the property that Q_in = Q_out*S, where S is an N-by-N diagonal matrix with diagonal entries equal to +1 or -1\&. The Householder vectors (columns V(i) of V) are stored in A on output, and the diagonal entries of S are stored in D\&. Block reflectors are also returned in T (same output format as SGEQRT)\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. M >= N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The column block size to be used in the reconstruction of Householder column vector blocks in the array A and corresponding block reflectors in the array T\&. NB >= 1\&. (Note that if NB > N, then N is used instead of NB as the column block size\&.) .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension (LDA,N) On entry: The array A contains an M-by-N orthonormal matrix Q_in, i\&.e the columns of A are orthogonal unit vectors\&. On exit: The elements below the diagonal of A represent the unit lower-trapezoidal matrix V of Householder column vectors V(i)\&. The unit diagonal entries of V are not stored (same format as the output below the diagonal in A from SGEQRT)\&. The matrix T and the matrix V stored on output in A implicitly define Q_out\&. The elements above the diagonal contain the factor U of the 'modified' LU-decomposition: Q_in - ( S ) = V * U ( 0 ) where 0 is a (M-N)-by-(M-N) zero matrix\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fIT\fP .PP .nf T is REAL array, dimension (LDT, N) Let NOCB = Number_of_output_col_blocks = CEIL(N/NB) On exit, T(1:NB, 1:N) contains NOCB upper-triangular block reflectors used to define Q_out stored in compact form as a sequence of upper-triangular NB-by-NB column blocks (same format as the output T in SGEQRT)\&. The matrix T and the matrix V stored on output in A implicitly define Q_out\&. NOTE: The lower triangles below the upper-triangular blocks will be filled with zeros\&. See Further Details\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,min(NB,N))\&. .fi .PP .br \fID\fP .PP .nf D is REAL array, dimension min(M,N)\&. The elements can be only plus or minus one\&. D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where 1 <= i <= min(M,N), and Q_in_i is Q_in after performing i-1 steps of “modified” Gaussian elimination\&. See Further Details\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf The computed M-by-M orthogonal factor Q_out is defined implicitly as a product of orthogonal matrices Q_out(i)\&. Each Q_out(i) is stored in the compact WY-representation format in the corresponding blocks of matrices V (stored in A) and T\&. The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N matrix A contains the column vectors V(i) in NB-size column blocks VB(j)\&. For example, VB(1) contains the columns V(1), V(2), \&.\&.\&. V(NB)\&. NOTE: The unit entries on the diagonal of Y are not stored in A\&. The number of column blocks is NOCB = Number_of_output_col_blocks = CEIL(N/NB) where each block is of order NB except for the last block, which is of order LAST_NB = N - (NOCB-1)*NB\&. For example, if M=6, N=5 and NB=2, the matrix V is V = ( VB(1), VB(2), VB(3) ) = = ( 1 ) ( v21 1 ) ( v31 v32 1 ) ( v41 v42 v43 1 ) ( v51 v52 v53 v54 1 ) ( v61 v62 v63 v54 v65 ) For each of the column blocks VB(i), an upper-triangular block reflector TB(i) is computed\&. These blocks are stored as a sequence of upper-triangular column blocks in the NB-by-N matrix T\&. The size of each TB(i) block is NB-by-NB, except for the last block, whose size is LAST_NB-by-LAST_NB\&. For example, if M=6, N=5 and NB=2, the matrix T is T = ( TB(1), TB(2), TB(3) ) = = ( t11 t12 t13 t14 t15 ) ( t22 t24 ) The M-by-M factor Q_out is given as a product of NOCB orthogonal M-by-M matrices Q_out(i)\&. Q_out = Q_out(1) * Q_out(2) * \&.\&.\&. * Q_out(NOCB), where each matrix Q_out(i) is given by the WY-representation using corresponding blocks from the matrices V and T: Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T, where I is the identity matrix\&. Here is the formula with matrix dimensions: Q(i){M-by-M} = I{M-by-M} - VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M}, where INB = NB, except for the last block NOCB for which INB=LAST_NB\&. ===== NOTE: ===== If Q_in is the result of doing a QR factorization B = Q_in * R_in, then: B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out\&. So if one wants to interpret Q_out as the result of the QR factorization of B, then the corresponding R_out should be equal to R_out = S * R_in, i\&.e\&. some rows of R_in should be multiplied by -1\&. For the details of the algorithm, see [1]\&. [1] 'Reconstructing Householder vectors from tall-skinny QR', G\&. Ballard, J\&. Demmel, L\&. Grigori, M\&. Jacquelin, H\&.D\&. Nguyen, E\&. Solomonik, J\&. Parallel Distrib\&. Comput\&., vol\&. 85, pp\&. 3-31, 2015\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley .fi .PP .RE .PP .SS "subroutine zunhr_col (integer m, integer n, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) d, integer info)" .PP \fBZUNHR_COL\fP .PP \fBPurpose:\fP .RS 4 .PP .nf ZUNHR_COL takes an M-by-N complex matrix Q_in with orthonormal columns as input, stored in A, and performs Householder Reconstruction (HR), i\&.e\&. reconstructs Householder vectors V(i) implicitly representing another M-by-N matrix Q_out, with the property that Q_in = Q_out*S, where S is an N-by-N diagonal matrix with diagonal entries equal to +1 or -1\&. The Householder vectors (columns V(i) of V) are stored in A on output, and the diagonal entries of S are stored in D\&. Block reflectors are also returned in T (same output format as ZGEQRT)\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. M >= N >= 0\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The column block size to be used in the reconstruction of Householder column vector blocks in the array A and corresponding block reflectors in the array T\&. NB >= 1\&. (Note that if NB > N, then N is used instead of NB as the column block size\&.) .fi .PP .br \fIA\fP .PP .nf A is COMPLEX*16 array, dimension (LDA,N) On entry: The array A contains an M-by-N orthonormal matrix Q_in, i\&.e the columns of A are orthogonal unit vectors\&. On exit: The elements below the diagonal of A represent the unit lower-trapezoidal matrix V of Householder column vectors V(i)\&. The unit diagonal entries of V are not stored (same format as the output below the diagonal in A from ZGEQRT)\&. The matrix T and the matrix V stored on output in A implicitly define Q_out\&. The elements above the diagonal contain the factor U of the 'modified' LU-decomposition: Q_in - ( S ) = V * U ( 0 ) where 0 is a (M-N)-by-(M-N) zero matrix\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fIT\fP .PP .nf T is COMPLEX*16 array, dimension (LDT, N) Let NOCB = Number_of_output_col_blocks = CEIL(N/NB) On exit, T(1:NB, 1:N) contains NOCB upper-triangular block reflectors used to define Q_out stored in compact form as a sequence of upper-triangular NB-by-NB column blocks (same format as the output T in ZGEQRT)\&. The matrix T and the matrix V stored on output in A implicitly define Q_out\&. NOTE: The lower triangles below the upper-triangular blocks will be filled with zeros\&. See Further Details\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,min(NB,N))\&. .fi .PP .br \fID\fP .PP .nf D is COMPLEX*16 array, dimension min(M,N)\&. The elements can be only plus or minus one\&. D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where 1 <= i <= min(M,N), and Q_in_i is Q_in after performing i-1 steps of “modified” Gaussian elimination\&. See Further Details\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf The computed M-by-M unitary factor Q_out is defined implicitly as a product of unitary matrices Q_out(i)\&. Each Q_out(i) is stored in the compact WY-representation format in the corresponding blocks of matrices V (stored in A) and T\&. The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N matrix A contains the column vectors V(i) in NB-size column blocks VB(j)\&. For example, VB(1) contains the columns V(1), V(2), \&.\&.\&. V(NB)\&. NOTE: The unit entries on the diagonal of Y are not stored in A\&. The number of column blocks is NOCB = Number_of_output_col_blocks = CEIL(N/NB) where each block is of order NB except for the last block, which is of order LAST_NB = N - (NOCB-1)*NB\&. For example, if M=6, N=5 and NB=2, the matrix V is V = ( VB(1), VB(2), VB(3) ) = = ( 1 ) ( v21 1 ) ( v31 v32 1 ) ( v41 v42 v43 1 ) ( v51 v52 v53 v54 1 ) ( v61 v62 v63 v54 v65 ) For each of the column blocks VB(i), an upper-triangular block reflector TB(i) is computed\&. These blocks are stored as a sequence of upper-triangular column blocks in the NB-by-N matrix T\&. The size of each TB(i) block is NB-by-NB, except for the last block, whose size is LAST_NB-by-LAST_NB\&. For example, if M=6, N=5 and NB=2, the matrix T is T = ( TB(1), TB(2), TB(3) ) = = ( t11 t12 t13 t14 t15 ) ( t22 t24 ) The M-by-M factor Q_out is given as a product of NOCB unitary M-by-M matrices Q_out(i)\&. Q_out = Q_out(1) * Q_out(2) * \&.\&.\&. * Q_out(NOCB), where each matrix Q_out(i) is given by the WY-representation using corresponding blocks from the matrices V and T: Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T, where I is the identity matrix\&. Here is the formula with matrix dimensions: Q(i){M-by-M} = I{M-by-M} - VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M}, where INB = NB, except for the last block NOCB for which INB=LAST_NB\&. ===== NOTE: ===== If Q_in is the result of doing a QR factorization B = Q_in * R_in, then: B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out\&. So if one wants to interpret Q_out as the result of the QR factorization of B, then the corresponding R_out should be equal to R_out = S * R_in, i\&.e\&. some rows of R_in should be multiplied by -1\&. For the details of the algorithm, see [1]\&. [1] 'Reconstructing Householder vectors from tall-skinny QR', G\&. Ballard, J\&. Demmel, L\&. Grigori, M\&. Jacquelin, H\&.D\&. Nguyen, E\&. Solomonik, J\&. Parallel Distrib\&. Comput\&., vol\&. 85, pp\&. 3-31, 2015\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 .PP .nf November 2019, Igor Kozachenko, Computer Science Division, University of California, Berkeley .fi .PP .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.