.TH CHPTRD 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME CHPTRD - a complex Hermitian matrix A stored in packed form to real symmetric tridiagonal form T by a unitary similarity transformation .SH SYNOPSIS .TP 19 SUBROUTINE CHPTRD( UPLO, N, AP, D, E, TAU, INFO ) .TP 19 .ti +4 CHARACTER UPLO .TP 19 .ti +4 INTEGER INFO, N .TP 19 .ti +4 REAL D( * ), E( * ) .TP 19 .ti +4 COMPLEX AP( * ), TAU( * ) .SH 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. .SH ARGUMENTS .TP 8 UPLO (input) CHARACTER*1 = \(aqU\(aq: Upper triangle of A is stored; .br = \(aqL\(aq: Lower triangle of A is stored. .TP 8 N (input) INTEGER The order of the matrix A. N >= 0. .TP 8 AP (input/output) 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 = \(aqU\(aq, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = \(aqL\(aq, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, if UPLO = \(aqU\(aq, 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 = \(aqL\(aq, 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 (output) REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i). .TP 8 E (output) REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = \(aqU\(aq, E(i) = A(i+1,i) if UPLO = \(aqL\(aq. .TP 8 TAU (output) COMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details). .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .SH FURTHER DETAILS If UPLO = \(aqU\(aq, the matrix Q is represented as a product of elementary reflectors .br Q = H(n-1) . . . H(2) H(1). .br Each H(i) has the form .br H(i) = I - tau * v * v\(aq .br 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). .br If UPLO = \(aqL\(aq, the matrix Q is represented as a product of elementary reflectors .br Q = H(1) H(2) . . . H(n-1). .br Each H(i) has the form .br H(i) = I - tau * v * v\(aq .br 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). .br