.TH "unhr_col" 3 "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 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] , !> 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 .PP Definition at line \fB258\fP of file \fBcunhr_col\&.f\fP\&. .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 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] , !> 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 .PP Definition at line \fB258\fP of file \fBdorhr_col\&.f\fP\&. .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 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] , !> 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 .PP Definition at line \fB258\fP of file \fBsorhr_col\&.f\fP\&. .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 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] , !> 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 .PP Definition at line \fB258\fP of file \fBzunhr_col\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.