hetrd(3)

NAME
hetrd - {he,sy}trd: reduction to tridiagonal

SYNOPSIS

  Functions
subroutine chetrd (uplo, n, a, lda, d, e, tau, work, lwork, info)
CHETRD
subroutine dsytrd (uplo, n, a, lda, d, e, tau, work, lwork, info)
DSYTRD
subroutine ssytrd (uplo, n, a, lda, d, e, tau, work, lwork, info)
SSYTRD
subroutine zhetrd (uplo, n, a, lda, d, e, tau, work, lwork, info)
ZHETRD

Detailed Description

Function Documentation

  subroutine chetrd (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)

CHETRD

Purpose:

    !>
    !> CHETRD reduces a complex Hermitian matrix A to real symmetric
    !> tridiagonal form T by a unitary similarity transformation:
    !> Q**H * A * Q = T.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    A

    !>          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.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          The diagonal elements of the tridiagonal matrix T:
    !>          D(i) = A(i,i).
    !>

    E

    !>          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'.
    !>

    TAU

    !>          TAU is COMPLEX array, dimension (N-1)
    !>          The scalar factors of the elementary reflectors (see Further
    !>          Details).
    !>

    WORK

    !>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.  LWORK >= 1.
    !>          For optimum performance LWORK >= N*NB, where NB is the
    !>          optimal blocksize.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  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).
    !>

Definition at line 191 of file chetrd.f.

  subroutine dsytrd (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) tau, double precision, dimension( * ) work, integer lwork, integer info)

DSYTRD

Purpose:

    !>
    !> DSYTRD reduces a real symmetric matrix A to real symmetric
    !> tridiagonal form T by an orthogonal similarity transformation:
    !> Q**T * A * Q = T.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    A

    !>          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.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          The diagonal elements of the tridiagonal matrix T:
    !>          D(i) = A(i,i).
    !>

    E

    !>          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'.
    !>

    TAU

    !>          TAU is DOUBLE PRECISION array, dimension (N-1)
    !>          The scalar factors of the elementary reflectors (see Further
    !>          Details).
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.  LWORK >= 1.
    !>          For optimum performance LWORK >= N*NB, where NB is the
    !>          optimal blocksize.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  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).
    !>

Definition at line 191 of file dsytrd.f.

  subroutine ssytrd (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau, real, dimension( * ) work, integer lwork, integer info)

SSYTRD

Purpose:

    !>
    !> SSYTRD reduces a real symmetric matrix A to real symmetric
    !> tridiagonal form T by an orthogonal similarity transformation:
    !> Q**T * A * Q = T.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    A

    !>          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.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    D

    !>          D is REAL array, dimension (N)
    !>          The diagonal elements of the tridiagonal matrix T:
    !>          D(i) = A(i,i).
    !>

    E

    !>          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'.
    !>

    TAU

    !>          TAU is REAL array, dimension (N-1)
    !>          The scalar factors of the elementary reflectors (see Further
    !>          Details).
    !>

    WORK

    !>          WORK is REAL array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.  LWORK >= 1.
    !>          For optimum performance LWORK >= N*NB, where NB is the
    !>          optimal blocksize.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  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).
    !>

Definition at line 191 of file ssytrd.f.

  subroutine zhetrd (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRD

Purpose:

    !>
    !> ZHETRD reduces a complex Hermitian matrix A to real symmetric
    !> tridiagonal form T by a unitary similarity transformation:
    !> Q**H * A * Q = T.
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangle of A is stored;
    !>          = 'L':  Lower triangle of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    A

    !>          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.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    D

    !>          D is DOUBLE PRECISION array, dimension (N)
    !>          The diagonal elements of the tridiagonal matrix T:
    !>          D(i) = A(i,i).
    !>

    E

    !>          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'.
    !>

    TAU

    !>          TAU is COMPLEX*16 array, dimension (N-1)
    !>          The scalar factors of the elementary reflectors (see Further
    !>          Details).
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.  LWORK >= 1.
    !>          For optimum performance LWORK >= N*NB, where NB is the
    !>          optimal blocksize.
    !>
    !>          If LWORK = -1, then a workspace query is assumed; the routine
    !>          only calculates the optimal size of the WORK array, returns
    !>          this value as the first entry of the WORK array, and no error
    !>          message related to LWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  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).
    !>

Definition at line 191 of file zhetrd.f.

Author

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