Provided by: liblapack-doc_3.12.0-3build1.1_all bug

NAME

       hptrd - {hp,sp}trd: reduction to tridiagonal

SYNOPSIS

   Functions
       subroutine chptrd (uplo, n, ap, d, e, tau, info)
           CHPTRD
       subroutine dsptrd (uplo, n, ap, d, e, tau, info)
           DSPTRD
       subroutine ssptrd (uplo, n, ap, d, e, tau, info)
           SSPTRD
       subroutine zhptrd (uplo, n, ap, d, e, tau, info)
           ZHPTRD

Detailed Description

Function Documentation

   subroutine chptrd (character uplo, integer n, complex, dimension( * ) ap, real, dimension( * )
       d, real, dimension( * ) e, complex, dimension( * ) tau, integer info)
       CHPTRD

       Purpose:

            CHPTRD reduces a complex Hermitian matrix A stored in packed form 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.

           AP

                     AP is COMPLEX array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
                     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.

           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).

           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 AP,
             overwriting A(1:i-1,i+1), and tau is stored 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 AP,
             overwriting A(i+2:n,i), and tau is stored in TAU(i).

   subroutine dsptrd (character uplo, integer n, double precision, dimension( * ) ap, double
       precision, dimension( * ) d, double precision, dimension( * ) e, double precision,
       dimension( * ) tau, integer info)
       DSPTRD

       Purpose:

            DSPTRD reduces a real symmetric matrix A stored in packed form to
            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.

           AP

                     AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
                     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.

           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).

           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 AP,
             overwriting A(1:i-1,i+1), and tau is stored 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 AP,
             overwriting A(i+2:n,i), and tau is stored in TAU(i).

   subroutine ssptrd (character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) d,
       real, dimension( * ) e, real, dimension( * ) tau, integer info)
       SSPTRD

       Purpose:

            SSPTRD reduces a real symmetric matrix A stored in packed form to
            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.

           AP

                     AP is REAL array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the symmetric matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
                     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.

           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).

           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 AP,
             overwriting A(1:i-1,i+1), and tau is stored 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 AP,
             overwriting A(i+2:n,i), and tau is stored in TAU(i).

   subroutine zhptrd (character uplo, integer n, complex*16, dimension( * ) ap, double precision,
       dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( * ) tau,
       integer info)
       ZHPTRD

       Purpose:

            ZHPTRD reduces a complex Hermitian matrix A stored in packed form 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.

           AP

                     AP is COMPLEX*16 array, dimension (N*(N+1)/2)
                     On entry, the upper or lower triangle of the Hermitian matrix
                     A, packed columnwise in a linear array.  The j-th column of A
                     is stored in the array AP as follows:
                     if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
                     if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
                     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.

           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).

           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 AP,
             overwriting A(1:i-1,i+1), and tau is stored 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 AP,
             overwriting A(i+2:n,i), and tau is stored in TAU(i).

Author

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