.TH "latrd" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME latrd \- latrd: step in hetrd .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBclatrd\fP (uplo, n, nb, a, lda, e, tau, w, ldw)" .br .RI "\fBCLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation\&. " .ti -1c .RI "subroutine \fBdlatrd\fP (uplo, n, nb, a, lda, e, tau, w, ldw)" .br .RI "\fBDLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation\&. " .ti -1c .RI "subroutine \fBslatrd\fP (uplo, n, nb, a, lda, e, tau, w, ldw)" .br .RI "\fBSLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation\&. " .ti -1c .RI "subroutine \fBzlatrd\fP (uplo, n, nb, a, lda, e, tau, w, ldw)" .br .RI "\fBZLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine clatrd (character uplo, integer n, integer nb, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) e, complex, dimension( * ) tau, complex, dimension( ldw, * ) w, integer ldw)" .PP \fBCLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation\&. .PP \fBPurpose:\fP .RS 4 .PP .nf CLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q**H * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A\&. If UPLO = 'U', CLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', CLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied\&. This is an auxiliary routine called by CHETRD\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of rows and columns to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX array, dimension (LDA,N) On entry, the Hermitian matrix A\&. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced\&. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced\&. On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors\&. 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 \fIE\fP .PP .nf E is REAL array, dimension (N-1) If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix\&. .fi .PP .br \fITAU\fP .PP .nf TAU is COMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'\&. See Further Details\&. .fi .PP .br \fIW\fP .PP .nf W is COMPLEX array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A\&. .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,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 If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n) H(n-1) \&. \&. \&. H(n-nb+1)\&. 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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1)\&. If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i)\&. The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W**H - W*V**H\&. The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: if UPLO = 'U': if UPLO = 'L': ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i)\&. .fi .PP .RE .PP .SS "subroutine dlatrd (character uplo, integer n, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, double precision, dimension( * ) tau, double precision, dimension( ldw, * ) w, integer ldw)" .PP \fBDLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation\&. .PP \fBPurpose:\fP .RS 4 .PP .nf DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q**T * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A\&. If UPLO = 'U', DLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied\&. This is an auxiliary routine called by DSYTRD\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of rows and columns to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A\&. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced\&. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced\&. On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors\&. See Further Details\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= (1,N)\&. .fi .PP .br \fIE\fP .PP .nf E is DOUBLE PRECISION array, dimension (N-1) If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix\&. .fi .PP .br \fITAU\fP .PP .nf TAU is DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'\&. See Further Details\&. .fi .PP .br \fIW\fP .PP .nf W is DOUBLE PRECISION array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A\&. .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,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 If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n) H(n-1) \&. \&. \&. H(n-nb+1)\&. 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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1)\&. If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i)\&. The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W**T - W*V**T\&. The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: if UPLO = 'U': if UPLO = 'L': ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i)\&. .fi .PP .RE .PP .SS "subroutine slatrd (character uplo, integer n, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, real, dimension( * ) tau, real, dimension( ldw, * ) w, integer ldw)" .PP \fBSLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation\&. .PP \fBPurpose:\fP .RS 4 .PP .nf SLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q**T * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A\&. If UPLO = 'U', SLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', SLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied\&. This is an auxiliary routine called by SSYTRD\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of rows and columns to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension (LDA,N) On entry, the symmetric matrix A\&. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced\&. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced\&. On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors\&. See Further Details\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= (1,N)\&. .fi .PP .br \fIE\fP .PP .nf E is REAL array, dimension (N-1) If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix\&. .fi .PP .br \fITAU\fP .PP .nf TAU is REAL array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'\&. See Further Details\&. .fi .PP .br \fIW\fP .PP .nf W is REAL array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A\&. .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,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 If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n) H(n-1) \&. \&. \&. H(n-nb+1)\&. 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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1)\&. If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i)\&. The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - V*W**T - W*V**T\&. The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: if UPLO = 'U': if UPLO = 'L': ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i)\&. .fi .PP .RE .PP .SS "subroutine zlatrd (character uplo, integer n, integer nb, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, complex*16, dimension( * ) tau, complex*16, dimension( ldw, * ) w, integer ldw)" .PP \fBZLATRD\fP reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation\&. .PP \fBPurpose:\fP .RS 4 .PP .nf ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q**H * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A\&. If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied\&. This is an auxiliary routine called by ZHETRD\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of rows and columns to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX*16 array, dimension (LDA,N) On entry, the Hermitian matrix A\&. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced\&. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced\&. On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors\&. 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 \fIE\fP .PP .nf E is DOUBLE PRECISION array, dimension (N-1) If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix\&. .fi .PP .br \fITAU\fP .PP .nf TAU is COMPLEX*16 array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'\&. See Further Details\&. .fi .PP .br \fIW\fP .PP .nf W is COMPLEX*16 array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A\&. .fi .PP .br \fILDW\fP .PP .nf LDW is INTEGER The leading dimension of the array W\&. LDW >= max(1,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 If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors Q = H(n) H(n-1) \&. \&. \&. H(n-nb+1)\&. 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(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1)\&. If UPLO = 'L', the matrix Q is represented as a product of 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) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i)\&. The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form: A := A - V*W**H - W*V**H\&. The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2: if UPLO = 'U': if UPLO = 'L': ( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a ) where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i)\&. .fi .PP .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.