Library Functions Manual

NAME¶

latrd - latrd: step in hetrd

SYNOPSIS¶

Functions¶

subroutine CLATRD (uplo, n, nb, a, lda, e, tau, w, ldw)
CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation. subroutine DLATRD (uplo, n, nb, a, lda, e, tau, w, ldw)
DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation. subroutine SLATRD (uplo, n, nb, a, lda, e, tau, w, ldw)
SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation. subroutine ZLATRD (uplo, n, nb, a, lda, e, tau, w, ldw)
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Detailed Description¶

Function Documentation¶

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)¶

CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Purpose:

!>
!> 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.
!>

Parameters

UPLO

!>          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
!>

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

!>          NB is INTEGER
!>          The number of rows and columns to be reduced.
!>

!>          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.
!>

LDA

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

!>          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.
!>

TAU

!>          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.
!>

!>          W is COMPLEX array, dimension (LDW,NB)
!>          The n-by-nb matrix W required to update the unreduced part
!>          of A.
!>

LDW

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

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) 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).
!>

Definition at line 198 of file clatrd.f.

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)¶

DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Purpose:

!>
!> 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.
!>

Parameters

UPLO

!>          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
!>

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

!>          NB is INTEGER
!>          The number of rows and columns to be reduced.
!>

!>          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.
!>

LDA

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

!>          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.
!>

TAU

!>          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.
!>

!>          W is DOUBLE PRECISION array, dimension (LDW,NB)
!>          The n-by-nb matrix W required to update the unreduced part
!>          of A.
!>

LDW

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

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) 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).
!>

Definition at line 197 of file dlatrd.f.

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)¶

SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.

Purpose:

!>
!> 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.
!>

Parameters

UPLO

!>          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
!>

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

!>          NB is INTEGER
!>          The number of rows and columns to be reduced.
!>

!>          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.
!>

LDA

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

!>          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.
!>

TAU

!>          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.
!>

!>          W is REAL array, dimension (LDW,NB)
!>          The n-by-nb matrix W required to update the unreduced part
!>          of A.
!>

LDW

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

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) 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).
!>

Definition at line 197 of file slatrd.f.

subroutine ZLATRD (character uplo, integer n, integer nb, complex16, dimension( lda, ) a, integer lda, double precision, dimension( * ) e, complex16, dimension( ) tau, complex16, dimension( ldw, ) w, integer ldw)¶

ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Purpose:

!>
!> 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.
!>

Parameters

UPLO

!>          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
!>

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

!>          NB is INTEGER
!>          The number of rows and columns to be reduced.
!>

!>          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.
!>

LDA

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

!>          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.
!>

TAU

!>          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.
!>

!>          W is COMPLEX*16 array, dimension (LDW,NB)
!>          The n-by-nb matrix W required to update the unreduced part
!>          of A.
!>

LDW

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

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) 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).
!>

Definition at line 198 of file zlatrd.f.

Author¶

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NAME¶

SYNOPSIS¶

Functions¶

Detailed Description¶

Function Documentation¶

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)¶

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)¶

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)¶

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)¶

Author¶

subroutine ZLATRD (character uplo, integer n, integer nb, complex16, dimension( lda, ) a, integer lda, double precision, dimension( * ) e, complex16, dimension( ) tau, complex16, dimension( ldw, ) w, integer ldw)¶