Hessenberg reduction: balancing, back-transformation, orthogonal factor generation. More...

Functions/Subroutines
pure subroutine, public	la_sgebak (job, side, n, ilo, ihi, scale, m, v, ldv, info)
	SGEBAK: forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by SGEBAL.

pure subroutine, public	la_dgebak (job, side, n, ilo, ihi, scale, m, v, ldv, info)
	DGEBAK: forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by DGEBAL.

pure subroutine, public	la_qgebak (job, side, n, ilo, ihi, scale, m, v, ldv, info)
	QGEBAK: forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by QGEBAL.

pure subroutine, public	la_sorghr (n, ilo, ihi, a, lda, tau, work, lwork, info)
	SORGHR: generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by SGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_dorghr (n, ilo, ihi, a, lda, tau, work, lwork, info)
	DORGHR: generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_qorghr (n, ilo, ihi, a, lda, tau, work, lwork, info)
	QORGHR: generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by QGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_sormhr (side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
	SORMHR: overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': Q*T C C * Q**T where Q is a real orthogonal matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by SGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_dormhr (side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
	DORMHR: overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': Q*T C C * Q**T where Q is a real orthogonal matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by DGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_qormhr (side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
	QORMHR: overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': Q*T C C * Q**T where Q is a real orthogonal matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by QGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_sgebal (job, n, a, lda, ilo, ihi, scale, info)
	SGEBAL: balances a general real matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

pure subroutine, public	la_dgebal (job, n, a, lda, ilo, ihi, scale, info)
	DGEBAL: balances a general real matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

pure subroutine, public	la_qgebal (job, n, a, lda, ilo, ihi, scale, info)
	QGEBAL: balances a general real matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

pure subroutine, public	la_sgehd2 (n, ilo, ihi, a, lda, tau, work, info)
	SGEHD2: reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q*T A * Q = H .

pure subroutine, public	la_dgehd2 (n, ilo, ihi, a, lda, tau, work, info)
	DGEHD2: reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q*T A * Q = H .

pure subroutine, public	la_qgehd2 (n, ilo, ihi, a, lda, tau, work, info)
	QGEHD2: reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q*T A * Q = H .

pure subroutine, public	la_slahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
	SLAHR2: reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q*T A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - VTV*T, and also the matrix Y = A V * T. This is an auxiliary routine called by SGEHRD.

pure subroutine, public	la_dlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
	DLAHR2: reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q*T A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - VTV*T, and also the matrix Y = A V * T. This is an auxiliary routine called by DGEHRD.

pure subroutine, public	la_qlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
	QLAHR2: reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q*T A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - VTV*T, and also the matrix Y = A V * T. This is an auxiliary routine called by QGEHRD.

pure subroutine, public	la_sgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
	SGEHRD: reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q*T A * Q = H .

pure subroutine, public	la_dgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
	DGEHRD: reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q*T A * Q = H .

pure subroutine, public	la_qgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
	QGEHRD: reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q*T A * Q = H .

pure subroutine, public	la_cgebak (job, side, n, ilo, ihi, scale, m, v, ldv, info)
	CGEBAK: forms the right or left eigenvectors of a complex general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by CGEBAL.

pure subroutine, public	la_zgebak (job, side, n, ilo, ihi, scale, m, v, ldv, info)
	ZGEBAK: forms the right or left eigenvectors of a complex general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by ZGEBAL.

pure subroutine, public	la_wgebak (job, side, n, ilo, ihi, scale, m, v, ldv, info)
	WGEBAK: forms the right or left eigenvectors of a complex general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by WGEBAL.

pure subroutine, public	la_cgebal (job, n, a, lda, ilo, ihi, scale, info)
	CGEBAL: balances a general complex matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

pure subroutine, public	la_zgebal (job, n, a, lda, ilo, ihi, scale, info)
	ZGEBAL: balances a general complex matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

pure subroutine, public	la_wgebal (job, n, a, lda, ilo, ihi, scale, info)
	WGEBAL: balances a general complex matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

pure subroutine, public	la_cgehd2 (n, ilo, ihi, a, lda, tau, work, info)
	CGEHD2: reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q*H A * Q = H .

pure subroutine, public	la_zgehd2 (n, ilo, ihi, a, lda, tau, work, info)
	ZGEHD2: reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q*H A * Q = H .

pure subroutine, public	la_wgehd2 (n, ilo, ihi, a, lda, tau, work, info)
	WGEHD2: reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q*H A * Q = H .

pure subroutine, public	la_clahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
	CLAHR2: reduces the first NB columns of A complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an unitary similarity transformation Q*H A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - VTv*H, and also the matrix Y = A V * T. This is an auxiliary routine called by CGEHRD.

pure subroutine, public	la_zlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
	ZLAHR2: reduces the first NB columns of A complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an unitary similarity transformation Q*H A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - VTV*H, and also the matrix Y = A V * T. This is an auxiliary routine called by ZGEHRD.

pure subroutine, public	la_wlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
	WLAHR2: reduces the first NB columns of A complex general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an unitary similarity transformation Q*H A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - VTV*H, and also the matrix Y = A V * T. This is an auxiliary routine called by WGEHRD.

pure subroutine, public	la_cunghr (n, ilo, ihi, a, lda, tau, work, lwork, info)
	CUNGHR: generates a complex unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by CGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_zunghr (n, ilo, ihi, a, lda, tau, work, lwork, info)
	ZUNGHR: generates a complex unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by ZGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_wunghr (n, ilo, ihi, a, lda, tau, work, lwork, info)
	WUNGHR: generates a complex unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by WGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_cunmhr (side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
	CUNMHR: overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q*H C C * Q**H where Q is a complex unitary matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by CGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_zunmhr (side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
	ZUNMHR: overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q*H C C * Q**H where Q is a complex unitary matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by ZGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_wunmhr (side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
	WUNMHR: overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q*H C C * Q**H where Q is a complex unitary matrix of order nq, with nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of IHI-ILO elementary reflectors, as returned by WGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

pure subroutine, public	la_cgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
	CGEHRD: reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation: Q*H A * Q = H .

pure subroutine, public	la_zgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
	ZGEHRD: reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation: Q*H A * Q = H .

pure subroutine, public	la_wgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
	WGEHRD: reduces a complex general matrix A to upper Hessenberg form H by an unitary similarity transformation: Q*H A * Q = H .

Detailed Description

Hessenberg reduction: balancing, back-transformation, orthogonal factor generation.

Function/Subroutine Documentation

◆ la_cgebak()

pure subroutine, public la_lapack_eigv_gen_hess::la_cgebak	(	character, intent(in)	job,
		character, intent(in)	side,
		integer(ilp), intent(in)	n,
		integer(ilp), intent(in)	ilo,
		integer(ilp), intent(in)	ihi,
		real(sp), dimension(*), intent(in)	scale,
		integer(ilp), intent(in)	m,
		complex(sp), dimension(ldv,*), intent(inout)	v,
		integer(ilp), intent(in)	ldv,
		integer(ilp), intent(out)	info )

CGEBAK: forms the right or left eigenvectors of a complex general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by CGEBAL.