.TH "lahr2" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME lahr2 \- lahr2: step in gehrd .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBclahr2\fP (n, k, nb, a, lda, tau, t, ldt, y, ldy)" .br .RI "\fBCLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. " .ti -1c .RI "subroutine \fBdlahr2\fP (n, k, nb, a, lda, tau, t, ldt, y, ldy)" .br .RI "\fBDLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. " .ti -1c .RI "subroutine \fBslahr2\fP (n, k, nb, a, lda, tau, t, ldt, y, ldy)" .br .RI "\fBSLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. " .ti -1c .RI "subroutine \fBzlahr2\fP (n, k, nb, a, lda, tau, t, ldt, y, ldy)" .br .RI "\fBZLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine clahr2 (integer n, integer k, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( nb ) tau, complex, dimension( ldt, nb ) t, integer ldt, complex, dimension( ldy, nb ) y, integer ldy)" .PP \fBCLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. .PP \fBPurpose:\fP .RS 4 .PP .nf CLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero\&. The reduction is performed by an unitary similarity transformation Q**H * A * Q\&. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*v**H, and also the matrix Y = A * V * T\&. This is an auxiliary routine called by CGEHRD\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fIK\fP .PP .nf K is INTEGER The offset for the reduction\&. Elements below the k-th subdiagonal in the first NB columns are reduced to zero\&. K < N\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of columns to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A\&. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors\&. The other columns of A are unchanged\&. See Further Details\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fITAU\fP .PP .nf TAU is COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors\&. See Further Details\&. .fi .PP .br \fIT\fP .PP .nf T is COMPLEX array, dimension (LDT,NB) The upper triangular matrix T\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= NB\&. .fi .PP .br \fIY\fP .PP .nf Y is COMPLEX array, dimension (LDY,NB) The n-by-nb matrix Y\&. .fi .PP .br \fILDY\fP .PP .nf LDY is INTEGER The leading dimension of the array Y\&. LDY >= N\&. .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 \fBFurther Details:\fP .RS 4 .PP .nf The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) \&. \&. \&. H(nb)\&. Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i)\&. The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - V*T*V**H) * (A - Y*V**H)\&. The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i)\&. This subroutine is a slight modification of LAPACK-3\&.0's CLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin\&. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3\&.0's CLAHRD routine\&. (This subroutine is not backward compatible with LAPACK-3\&.0's CLAHRD\&.) .fi .PP .RE .PP \fBReferences:\fP .RS 4 Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006\&. .RE .PP .SS "subroutine dlahr2 (integer n, integer k, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( nb ) tau, double precision, dimension( ldt, nb ) t, integer ldt, double precision, dimension( ldy, nb ) y, integer ldy)" .PP \fBDLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. .PP \fBPurpose:\fP .RS 4 .PP .nf DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero\&. The reduction is performed by an orthogonal similarity transformation Q**T * A * Q\&. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T\&. This is an auxiliary routine called by DGEHRD\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fIK\fP .PP .nf K is INTEGER The offset for the reduction\&. Elements below the k-th subdiagonal in the first NB columns are reduced to zero\&. K < N\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of columns to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is DOUBLE PRECISION array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A\&. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors\&. The other columns of A are unchanged\&. See Further Details\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fITAU\fP .PP .nf TAU is DOUBLE PRECISION array, dimension (NB) The scalar factors of the elementary reflectors\&. See Further Details\&. .fi .PP .br \fIT\fP .PP .nf T is DOUBLE PRECISION array, dimension (LDT,NB) The upper triangular matrix T\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= NB\&. .fi .PP .br \fIY\fP .PP .nf Y is DOUBLE PRECISION array, dimension (LDY,NB) The n-by-nb matrix Y\&. .fi .PP .br \fILDY\fP .PP .nf LDY is INTEGER The leading dimension of the array Y\&. LDY >= N\&. .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 \fBFurther Details:\fP .RS 4 .PP .nf The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) \&. \&. \&. H(nb)\&. Each H(i) has the form H(i) = I - tau * v * v**T where tau is a real scalar, and v is a real vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i)\&. The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - V*T*V**T) * (A - Y*V**T)\&. The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i)\&. This subroutine is a slight modification of LAPACK-3\&.0's DLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin\&. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3\&.0's DLAHRD routine\&. (This subroutine is not backward compatible with LAPACK-3\&.0's DLAHRD\&.) .fi .PP .RE .PP \fBReferences:\fP .RS 4 Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006\&. .RE .PP .SS "subroutine slahr2 (integer n, integer k, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( nb ) tau, real, dimension( ldt, nb ) t, integer ldt, real, dimension( ldy, nb ) y, integer ldy)" .PP \fBSLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. .PP \fBPurpose:\fP .RS 4 .PP .nf SLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero\&. The reduction is performed by an orthogonal similarity transformation Q**T * A * Q\&. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T\&. This is an auxiliary routine called by SGEHRD\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fIK\fP .PP .nf K is INTEGER The offset for the reduction\&. Elements below the k-th subdiagonal in the first NB columns are reduced to zero\&. K < N\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of columns to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A\&. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors\&. The other columns of A are unchanged\&. See Further Details\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fITAU\fP .PP .nf TAU is REAL array, dimension (NB) The scalar factors of the elementary reflectors\&. See Further Details\&. .fi .PP .br \fIT\fP .PP .nf T is REAL array, dimension (LDT,NB) The upper triangular matrix T\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= NB\&. .fi .PP .br \fIY\fP .PP .nf Y is REAL array, dimension (LDY,NB) The n-by-nb matrix Y\&. .fi .PP .br \fILDY\fP .PP .nf LDY is INTEGER The leading dimension of the array Y\&. LDY >= N\&. .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 \fBFurther Details:\fP .RS 4 .PP .nf The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) \&. \&. \&. H(nb)\&. Each H(i) has the form H(i) = I - tau * v * v**T where tau is a real scalar, and v is a real vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i)\&. The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - V*T*V**T) * (A - Y*V**T)\&. The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i)\&. This subroutine is a slight modification of LAPACK-3\&.0's SLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin\&. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3\&.0's SLAHRD routine\&. (This subroutine is not backward compatible with LAPACK-3\&.0's SLAHRD\&.) .fi .PP .RE .PP \fBReferences:\fP .RS 4 Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006\&. .RE .PP .SS "subroutine zlahr2 (integer n, integer k, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( nb ) tau, complex*16, dimension( ldt, nb ) t, integer ldt, complex*16, dimension( ldy, nb ) y, integer ldy)" .PP \fBZLAHR2\fP reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A\&. .PP \fBPurpose:\fP .RS 4 .PP .nf ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero\&. The reduction is performed by an unitary similarity transformation Q**H * A * Q\&. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T\&. This is an auxiliary routine called by ZGEHRD\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fIK\fP .PP .nf K is INTEGER The offset for the reduction\&. Elements below the k-th subdiagonal in the first NB columns are reduced to zero\&. K < N\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of columns to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX*16 array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A\&. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors\&. The other columns of A are unchanged\&. See Further Details\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fITAU\fP .PP .nf TAU is COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors\&. See Further Details\&. .fi .PP .br \fIT\fP .PP .nf T is COMPLEX*16 array, dimension (LDT,NB) The upper triangular matrix T\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= NB\&. .fi .PP .br \fIY\fP .PP .nf Y is COMPLEX*16 array, dimension (LDY,NB) The n-by-nb matrix Y\&. .fi .PP .br \fILDY\fP .PP .nf LDY is INTEGER The leading dimension of the array Y\&. LDY >= N\&. .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 \fBFurther Details:\fP .RS 4 .PP .nf The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) \&. \&. \&. H(nb)\&. Each H(i) has the form H(i) = I - tau * v * v**H where tau is a complex scalar, and v is a complex vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i)\&. The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - V*T*V**H) * (A - Y*V**H)\&. The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i)\&. This subroutine is a slight modification of LAPACK-3\&.0's ZLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin\&. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3\&.0's ZLAHRD routine\&. (This subroutine is not backward compatible with LAPACK-3\&.0's ZLAHRD\&.) .fi .PP .RE .PP \fBReferences:\fP .RS 4 Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006\&. .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.