hbtrd(3)

NAME
hbtrd - {hb,sb}trd: reduction to tridiagonal

SYNOPSIS

  Functions

subroutine chbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
CHBTRD
subroutine dsbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
DSBTRD
subroutine ssbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
SSBTRD
subroutine zhbtrd (vect, uplo, n, kd, ab, ldab, d, e, q, ldq, work, info)
ZHBTRD

Detailed Description

Function Documentation

  subroutine chbtrd (character vect, character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * ) work, integer info)

CHBTRD

Purpose:

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

Parameters
    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  do not form Q;
    !>          = 'V':  form Q;
    !>          = 'U':  update a matrix X, by forming X*Q.
    !>

    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.
    !>

    KD

    !>          KD is INTEGER
    !>          The number of superdiagonals of the matrix A if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>

    AB

    !>          AB is COMPLEX array, dimension (LDAB,N)
    !>          On entry, the upper or lower triangle of the Hermitian band
    !>          matrix A, stored in the first KD+1 rows of the array.  The
    !>          j-th column of A is stored in the j-th column of the array AB
    !>          as follows:
    !>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    !>          On exit, the diagonal elements of AB are overwritten by the
    !>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
    !>          elements on the first superdiagonal (if UPLO = 'U') or the
    !>          first subdiagonal (if UPLO = 'L') are overwritten by the
    !>          off-diagonal elements of T; the rest of AB is overwritten by
    !>          values generated during the reduction.
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= KD+1.
    !>

    D

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

    E

    !>          E is REAL array, dimension (N-1)
    !>          The off-diagonal elements of the tridiagonal matrix T:
    !>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
    !>

    Q

    !>          Q is COMPLEX array, dimension (LDQ,N)
    !>          On entry, if VECT = 'U', then Q must contain an N-by-N
    !>          matrix X; if VECT = 'N' or 'V', then Q need not be set.
    !>
    !>          On exit:
    !>          if VECT = 'V', Q contains the N-by-N unitary matrix Q;
    !>          if VECT = 'U', Q contains the product X*Q;
    !>          if VECT = 'N', the array Q is not referenced.
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q.
    !>          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
    !>

    WORK

    !>          WORK is COMPLEX array, dimension (N)
    !>

    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:

    !>
    !>  Modified by Linda Kaufman, Bell Labs.
    !>

  subroutine dsbtrd (character vect, character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( * ) work, integer info)

DSBTRD

Purpose:

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

Parameters
    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  do not form Q;
    !>          = 'V':  form Q;
    !>          = 'U':  update a matrix X, by forming X*Q.
    !>

    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.
    !>

    KD

    !>          KD is INTEGER
    !>          The number of superdiagonals of the matrix A if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>

    AB

    !>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
    !>          On entry, the upper or lower triangle of the symmetric band
    !>          matrix A, stored in the first KD+1 rows of the array.  The
    !>          j-th column of A is stored in the j-th column of the array AB
    !>          as follows:
    !>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    !>          On exit, the diagonal elements of AB are overwritten by the
    !>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
    !>          elements on the first superdiagonal (if UPLO = 'U') or the
    !>          first subdiagonal (if UPLO = 'L') are overwritten by the
    !>          off-diagonal elements of T; the rest of AB is overwritten by
    !>          values generated during the reduction.
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= KD+1.
    !>

    D

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

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          The off-diagonal elements of the tridiagonal matrix T:
    !>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
    !>

    Q

    !>          Q is DOUBLE PRECISION array, dimension (LDQ,N)
    !>          On entry, if VECT = 'U', then Q must contain an N-by-N
    !>          matrix X; if VECT = 'N' or 'V', then Q need not be set.
    !>
    !>          On exit:
    !>          if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
    !>          if VECT = 'U', Q contains the product X*Q;
    !>          if VECT = 'N', the array Q is not referenced.
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q.
    !>          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (N)
    !>

    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:

    !>
    !>  Modified by Linda Kaufman, Bell Labs.
    !>

  subroutine ssbtrd (character vect, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer info)

SSBTRD

Purpose:

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

Parameters
    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  do not form Q;
    !>          = 'V':  form Q;
    !>          = 'U':  update a matrix X, by forming X*Q.
    !>

    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.
    !>

    KD

    !>          KD is INTEGER
    !>          The number of superdiagonals of the matrix A if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>

    AB

    !>          AB is REAL array, dimension (LDAB,N)
    !>          On entry, the upper or lower triangle of the symmetric band
    !>          matrix A, stored in the first KD+1 rows of the array.  The
    !>          j-th column of A is stored in the j-th column of the array AB
    !>          as follows:
    !>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    !>          On exit, the diagonal elements of AB are overwritten by the
    !>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
    !>          elements on the first superdiagonal (if UPLO = 'U') or the
    !>          first subdiagonal (if UPLO = 'L') are overwritten by the
    !>          off-diagonal elements of T; the rest of AB is overwritten by
    !>          values generated during the reduction.
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= KD+1.
    !>

    D

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

    E

    !>          E is REAL array, dimension (N-1)
    !>          The off-diagonal elements of the tridiagonal matrix T:
    !>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
    !>

    Q

    !>          Q is REAL array, dimension (LDQ,N)
    !>          On entry, if VECT = 'U', then Q must contain an N-by-N
    !>          matrix X; if VECT = 'N' or 'V', then Q need not be set.
    !>
    !>          On exit:
    !>          if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
    !>          if VECT = 'U', Q contains the product X*Q;
    !>          if VECT = 'N', the array Q is not referenced.
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q.
    !>          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
    !>

    WORK

    !>          WORK is REAL array, dimension (N)
    !>

    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:

    !>
    !>  Modified by Linda Kaufman, Bell Labs.
    !>

  subroutine zhbtrd (character vect, character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( * ) work, integer info)

ZHBTRD

Purpose:

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

Parameters
    VECT

    !>          VECT is CHARACTER*1
    !>          = 'N':  do not form Q;
    !>          = 'V':  form Q;
    !>          = 'U':  update a matrix X, by forming X*Q.
    !>

    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.
    !>

    KD

    !>          KD is INTEGER
    !>          The number of superdiagonals of the matrix A if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>

    AB

    !>          AB is COMPLEX*16 array, dimension (LDAB,N)
    !>          On entry, the upper or lower triangle of the Hermitian band
    !>          matrix A, stored in the first KD+1 rows of the array.  The
    !>          j-th column of A is stored in the j-th column of the array AB
    !>          as follows:
    !>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    !>          On exit, the diagonal elements of AB are overwritten by the
    !>          diagonal elements of the tridiagonal matrix T; if KD > 0, the
    !>          elements on the first superdiagonal (if UPLO = 'U') or the
    !>          first subdiagonal (if UPLO = 'L') are overwritten by the
    !>          off-diagonal elements of T; the rest of AB is overwritten by
    !>          values generated during the reduction.
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= KD+1.
    !>

    D

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

    E

    !>          E is DOUBLE PRECISION array, dimension (N-1)
    !>          The off-diagonal elements of the tridiagonal matrix T:
    !>          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.
    !>

    Q

    !>          Q is COMPLEX*16 array, dimension (LDQ,N)
    !>          On entry, if VECT = 'U', then Q must contain an N-by-N
    !>          matrix X; if VECT = 'N' or 'V', then Q need not be set.
    !>
    !>          On exit:
    !>          if VECT = 'V', Q contains the N-by-N unitary matrix Q;
    !>          if VECT = 'U', Q contains the product X*Q;
    !>          if VECT = 'N', the array Q is not referenced.
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q.
    !>          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (N)
    !>

    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:

    !>
    !>  Modified by Linda Kaufman, Bell Labs.
    !>

Author

Generated automatically by Doxygen for LAPACK from the source code.