.TH "hetd2" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME hetd2 \- {he,sy}td2: reduction to tridiagonal, level 2 .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBchetd2\fP (uplo, n, a, lda, d, e, tau, info)" .br .RI "\fBCHETD2\fP reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm)\&. " .ti -1c .RI "subroutine \fBdsytd2\fP (uplo, n, a, lda, d, e, tau, info)" .br .RI "\fBDSYTD2\fP reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm)\&. " .ti -1c .RI "subroutine \fBssytd2\fP (uplo, n, a, lda, d, e, tau, info)" .br .RI "\fBSSYTD2\fP reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm)\&. " .ti -1c .RI "subroutine \fBzhetd2\fP (uplo, n, a, lda, d, e, tau, info)" .br .RI "\fBZHETD2\fP reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm)\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine chetd2 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tau, integer info)" .PP \fBCHETD2\fP reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm)\&. .PP \fBPurpose:\fP .RS 4 .PP .nf CHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T\&. .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\&. N >= 0\&. .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 diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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 \fID\fP .PP .nf D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i)\&. .fi .PP .br \fIE\fP .PP .nf E is REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'\&. .fi .PP .br \fITAU\fP .PP .nf TAU is COMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors (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 \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-1) \&. \&. \&. H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i)\&. If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) \&. \&. \&. H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i)\&. The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i)\&. .fi .PP .RE .PP .SS "subroutine dsytd2 (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tau, integer info)" .PP \fBDSYTD2\fP reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm)\&. .PP \fBPurpose:\fP .RS 4 .PP .nf DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T\&. .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\&. N >= 0\&. .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 diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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 >= max(1,N)\&. .fi .PP .br \fID\fP .PP .nf D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i)\&. .fi .PP .br \fIE\fP .PP .nf E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'\&. .fi .PP .br \fITAU\fP .PP .nf TAU is DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors (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 \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-1) \&. \&. \&. H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i)\&. If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) \&. \&. \&. H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i)\&. The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i)\&. .fi .PP .RE .PP .SS "subroutine ssytd2 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, integer info)" .PP \fBSSYTD2\fP reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm)\&. .PP \fBPurpose:\fP .RS 4 .PP .nf SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T\&. .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\&. N >= 0\&. .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 diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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 >= max(1,N)\&. .fi .PP .br \fID\fP .PP .nf D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i)\&. .fi .PP .br \fIE\fP .PP .nf E is REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'\&. .fi .PP .br \fITAU\fP .PP .nf TAU is REAL array, dimension (N-1) The scalar factors of the elementary reflectors (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 \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-1) \&. \&. \&. H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i)\&. If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) \&. \&. \&. H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i)\&. The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i)\&. .fi .PP .RE .PP .SS "subroutine zhetd2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, integer info)" .PP \fBZHETD2\fP reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm)\&. .PP \fBPurpose:\fP .RS 4 .PP .nf ZHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T\&. .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\&. N >= 0\&. .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 diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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 \fID\fP .PP .nf D is DOUBLE PRECISION array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i)\&. .fi .PP .br \fIE\fP .PP .nf E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'\&. .fi .PP .br \fITAU\fP .PP .nf TAU is COMPLEX*16 array, dimension (N-1) The scalar factors of the elementary reflectors (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 \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-1) \&. \&. \&. H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i)\&. If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) \&. \&. \&. H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i)\&. The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v2 v3 v4 ) ( d ) ( d e v3 v4 ) ( e d ) ( d e v4 ) ( v1 e d ) ( d e ) ( v1 v2 e d ) ( d ) ( v1 v2 v3 e d ) where d and e denote diagonal and off-diagonal elements of T, 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\&.