Module_Hous_Procedures¶

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MODULE EXPORTING HOUSEHOLDER REFLECTIONS.

LATEST REVISION : 16/09/2020

`subroutine hous1 ( u, tau )`¶

Purpose¶

HOUS1 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I and D’ = ( beta 0 )

where beta is scalar and X is an n-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is an n-element real vector with v(1) = 1.

If the elements of X(2:n) are all zero or size(X)=1, then tau = 0 and H is taken to be the unit matrix.

Otherwise 1 <= tau <= 2.

Arguments¶

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the vector X.

On exit, it is overwritten with the vector [beta v(2:n) ].

TAU (OUTPUT) real(stnd)

On exit, the value tau.

Further Details¶

This subroutine is based on the routine DLARFG in LAPACK77 (version 3) with improvements suggested by E. Anderson and M. Fahey. See,

Anderson, E., and Fahey, M., 1997:

Performance improvements to LAPACK for the Cray Scientific Library. LAPACK Working Note No 126.

`subroutine hous1 ( u, tau, beta )`¶

Purpose¶

HOUS1 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I and D’ = ( beta 0 )

where beta is scalar and X is an n-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is an n-element real vector with v(1) = 1.

If the elements of X(2:n) are all zero or size(X)=1, then tau = 0 and H is taken to be the unit matrix.

Otherwise 1 <= tau <= 2.

Arguments¶

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the vector x.

On exit, it is overwritten with the vector [ 1 v(2:n) ].

TAU (OUTPUT) real(stnd)

On exit, the value tau.

BETA (OUTPUT) real(stnd)

On exit, the value beta.

Further Details¶

This subroutine is based on the routine DLARFG in LAPACK77 (version 3) with improvements suggested by E. Anderson and M. Fahey. See:

Anderson, E., and Fahey, M., 1997:

Performance improvements to LAPACK for the Cray Scientific Library. LAPACK Working Note No 126.

`subroutine hous2 ( pivot, u, tau )`¶

Purpose¶

HOUS2 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I , X’ = ( alpha x ) and D’ = ( beta 0 )

where alpha and beta are scalars, and x is an (n-1)-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is an n-element real vector with v(1) = 1.

If the elements of x are all zero, then tau = 0 and H is taken to be the unit matrix.

Otherwise 1 <= tau <= 2.

Arguments¶

PIVOT (INPUT/OUTPUT) real(stnd)

On entry, the value alpha. On exit, it is overwritten with the value beta.

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the (n-1)-element vector x.

On exit, it is overwritten with the vector v(2:n) .

TAU (OUTPUT) real(stnd)

On exit, the value tau.

Further Details¶

This subroutine is based on the routine DLARFG in LAPACK77 (version 3) with improvements suggested by E. Anderson and M. Fahey. See:

Anderson, E., and Fahey, M., 1997:

Performance improvements to LAPACK for the Cray Scientific Library. LAPACK Working Note No 126.

`subroutine apply_hous1 ( u, tau, vec )`¶

Purpose¶

APPLY_HOUS1 applies a real elementary reflector H generated by HOUS1 to a real vector C. H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is an n-element real vector with v(1) = 1.

If tau = 0, then H is taken to be the unit matrix.

Arguments¶

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, U(2:) contains the vector v(2:) in the representation of H as output by HOUS1. U is not used if tau = 0.

U is restored on exit.

TAU (INPUT) real(stnd)

The value tau in the representation of H as output by HOUS1.

VEC (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the real vector C.

On exit, C is overwritten by the vector H * C

Further Details¶

It is assumed that size( VEC )>=size( U ) .

`subroutine apply_hous1 ( u, tau, mat, left )`¶

Purpose¶

APPLY_HOUS1 applies a real elementary reflector H generated by HOUS1 to a real n by m or m by n matrix, C, from the left or the right. H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is an n-element real vector with v(1) = 1.

If tau = 0, then H is taken to be the unit matrix.

Arguments¶

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, U(2:) contains the vector v(2:) in the representation of H as output by HOUS1. U is not used if tau = 0.

U is restored on exit.

TAU (INPUT) real(stnd)

The value tau in the representation of H as output by HOUS1.

MAT (INPUT/OUTPUT) real(stnd), dimension(:,:)

On entry, the n by m or m by n matrix C.

On exit, C is overwritten by the matrix H * C (LEFT=true) or C * H (LEFT=false).

LEFT (INPUT) logical(lgl)

If:

LEFT=true, H is applied to the real matrix C from the left

LEFT=false, H is applied to the real matrix C from the right.

Further Details¶

It is assumed that:

size( MAT, 1 ) >= size( U ) if LEFT=true

size( MAT, 2 ) >= size( U ) if LEFT=false.

`subroutine apply_hous2 ( u, tau, piv, vec )`¶

Purpose¶

APPLY_HOUS2 applies a real n-by-n elementary reflector H generated by HOUS2 to a real vector C. H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is an n-element real vector with v(1) = 1.

If tau = 0, then H is taken to be the unit matrix.

Arguments¶

U (INPUT) real(stnd), dimension(:)

On entry, U contains the vector v(2:n) in the representation of H as output by HOUS2. U is not used if tau = 0.

TAU (INPUT) real(stnd)

The value tau in the representation of H as output by HOUS2.

PIV (INPUT/OUTPUT) real(stnd)

On entry, the scalar C[1].

On exit, PIV is overwritten by the scalar (H * C)[1].

VEC (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the real vector C[2:].

On exit, C is overwritten by the vector (H * C)[2:].

Further Details¶

It is assumed that size( VEC ) >= size( U ).

`subroutine apply_hous2 ( u, tau, vec_piv, mat, left )`¶

Purpose¶

APPLY_HOUS2 applies a real n-by-n elementary reflector H generated by HOUS2 to a real n by m or m by n matrix, C, from the left or the right. H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is an n-element real vector with v(1) = 1.

If tau = 0, then H is taken to be the unit matrix.

Arguments¶

U (INPUT) real(stnd), dimension(:)

On entry, U contains the vector v(2:n) in the representation of H as output by HOUS2. U is not used if tau = 0.

TAU (INPUT) real(stnd)

The value tau in the representation of H as output by HOUS2.

VEC_PIV (INPUT/OUTPUT) real(stnd), dimension(:)

If LEFT=true:

On entry, the row_vector C[1,:]

On exit, VEC_PIV is overwritten by the vector (H * C)[1,:].

If LEFT=false:

On entry, the column_vector C[:,1]

On exit, VEC_PIV is overwritten by the vector (C * H)[:,1].

MAT (INPUT/OUTPUT) real(stnd), dimension(:,:)

If LEFT=true:

On entry, the n-1 by m matrix C[2:,:]

On exit, MAT is overwritten by the matrix (H * C)[2:,:].

If LEFT=false:

On entry, the m by n-1 matrix C[:,2:].

On exit, MAT is overwritten by the matrix (C * H)[:,2:].

LEFT (INPUT) logical(lgl)

If:

LEFT=true, H is applied to the real matrix C from the left

LEFT=false, H is applied to the real matrix C from the right.

Further Details¶

It is assumed that:

size( MAT, 1 ) >= size( U ) and size( MAT, 2 ) >= size( VEC_PIV ) if LEFT=true

size( MAT, 2 ) >= size( U ) and size( MAT, 1 ) >= size( VEC_PIV ) if LEFT=false.

`subroutine h1 ( u, beta, tau )`¶

Purpose¶

H1 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I and D’ = ( beta 0 )

where beta is scalar and X is an n-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

Arguments¶

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, U contains the pivot vector X.

On exit, U contains the vector v of the Householder reflector.

BETA (OUTPUT) real(stnd)

On exit, the value beta.

TAU (OUTPUT) real(stnd)

On exit, the value tau.

Further Details¶

On output, H is the identity matrix if TAU = 0.

This subroutine is adapted from

Lawson, C.L., and Hanson, R.J., 1974: Solving least square problems.

Prentice-Hall.

`subroutine h1 ( u, beta, tau, vec )`¶

Purpose¶

H1 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I and D’ = ( beta 0 )

where beta is scalar and X is an n-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

The real elementary reflector H is then applied to a real vector C .

Arguments¶

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, U contains the pivot vector X.

On exit, U contains the vector v of the Householder reflector.

BETA (OUTPUT) real(stnd)

On exit, the value beta.

TAU (OUTPUT) real(stnd)

On exit, the value tau.

VEC (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the real vector C .

On exit, C is overwritten by the vector H * C .

Further Details¶

On output, H is the identity matrix if TAU = 0.

It is assumed that size( VEC ) >= size( U ) = n .

This subroutine is adapted from:

Lawson, C.L., and Hanson, R.J., 1974:

Solving least square problems. Prentice-Hall.

`subroutine h1 ( u, beta, tau, mat, left )`¶

Purpose¶

H1 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I and D’ = ( beta 0 )

where beta is scalar and X is an n-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

The real elementary reflector H is then applied to a real matrix C from the left or the right.

Arguments¶

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, U contains the pivot vector X.

On exit, U contains the vector v of the Householder reflector.

BETA (OUTPUT) real(stnd)

On exit, the value beta.

TAU (OUTPUT) real(stnd)

On exit, the value tau.

MAT (INPUT/OUTPUT) real(stnd), dimension(:,:)

On entry, the n by m or m by n real matrix C.

On exit, C is overwritten by the matrix H * C (if LEFT=true) or C * H (if LEFT=false).

LEFT (INPUT) logical(lgl)

If:

LEFT=true, H is applied to the real matrix C from the left

LEFT=false, H is applied to the real matrix C from the right.

Further Details¶

On output, H is the identity matrix if TAU = 0.

It is assumed that:

size( MAT, 1 ) >= size( U ) if LEFT=true

size( MAT, 2 ) >= size( U ) if LEFT=false.

This subroutine is adapted from:

Lawson, C.L., and Hanson, R.J., 1974:

Solving least square problems. Prentice-Hall.

`subroutine h2 ( beta, u, up, tau )`¶

Purpose¶

H2 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I , X’ = ( alpha x ) and D’ = ( beta 0 )

where alpha and beta are scalars, and x is an (n-1)-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

Arguments¶

BETA (INPUT/OUTPUT) real(stnd)

On entry, the value alpha.

On exit, it is overwritten with the value beta.

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, U contains the pivot (n-1)-element vector x.

On exit, U contains the vector v(2:) of the Householder reflector.

UP (OUTPUT) real(stnd)

On exit, UP contains the value v(1) of the Householder reflector.

TAU (OUTPUT) real(stnd)

On exit, TAU contains the value tau of the Householder reflector.

Further Details¶

On output, H is the identity matrix if TAU = 0.

This subroutine is adapted from:

Lawson, C.L., and Hanson, R.J., 1974:

Solving least square problems. Prentice-Hall.

`subroutine h2 ( beta, u, up, tau, piv, vec )`¶

Purpose¶

H2 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I , X’ = ( alpha x ) and D’ = ( beta 0 )

where alpha and beta are scalars, and x is an (n-1)-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

The real elementary reflector H is then applied to a real vector C .

Arguments¶

BETA (INPUT/OUTPUT) real(stnd)

On entry, the value alpha.

On exit, it is overwritten with the value beta.

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, U contains the pivot (n-1)-element vector x. On exit, U contains the vector v(2:) of the Householder reflector.

UP (OUTPUT) real(stnd)

On exit, UP contains the value v(1) of the Householder reflector.

TAU (OUTPUT) real(stnd)

On exit, TAU contains the value tau of the Householder reflector.

PIV (INPUT/OUTPUT) real(stnd)

On entry, the scalar C[1].

On exit, PIV is overwritten by the scalar (H * C)[1].

VEC (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the real vector C[2:].

On exit, C is overwritten by the vector (H * C)[2:].

Further Details¶

On output, H is the identity matrix if TAU = 0.

It is assumed that size( VEC ) >= size( U ) .

This subroutine is adapted from:

Lawson, C.L., and Hanson, R.J., 1974:

Solving least square problems. Prentice-Hall.

`subroutine h2 ( beta, u, up, tau, vec_piv, mat, left )`¶

Purpose¶

H2 generates a real elementary reflector H of order n, such that

H * X = D , with H’ * H = I , X’ = ( alpha x ) and D’ = ( beta 0 )

where alpha and beta are scalars, and x is an (n-1)-element real vector.

H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

The real elementary reflector H is then applied to a real matrix C from the left or the right.

Arguments¶

BETA (INPUT/OUTPUT) real(stnd)

On entry, the value alpha.

On exit, it is overwritten with the value beta.

U (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, U contains the pivot (n-1)-element vector x. On exit, U contains the vector v(2:) of the Householder reflector.

UP (OUTPUT) real(stnd)

On exit, UP contains the value v(1) of the Householder reflector.

TAU (OUTPUT) real(stnd)

On exit, TAU contains the value tau of the Householder reflector.

VEC_PIV (INPUT/OUTPUT) real(stnd), dimension(:)

If LEFT=true:

On entry, the row_vector C[1,:]

On exit, VEC_PIV is overwritten by the vector (H * C)[1,:].

If LEFT=false:

On entry, the column_vector C[:,1]

On exit, VEC_PIV is overwritten by the vector (C * H)[:,1].

MAT (INPUT/OUTPUT) real(stnd), dimension(:,:)

If LEFT=true:

On entry, the n-1 by m matrix C[2:,:]

On exit, MAT is overwritten by the matrix (H * C)[2:,:] .

If LEFT=false:

On entry, the m by n-1 matrix C[:,2:]

On exit, MAT is overwritten by the matrix (C * H)[:,2:].

LEFT (INPUT) logical(lgl)

If:

LEFT=true, H is applied to the real matrix C from the left

LEFT=false, H is applied to the real matrix C from the right.

Further Details¶

On output, H is the identity matrix if TAU = 0.

It is assumed that:

size( MAT, 1 ) >= size( U ) and size( MAT, 2 ) >= size( VEC_PIV ) if LEFT=true.

size( MAT, 2 ) >= size( U ) and size( MAT, 1 ) >= size( VEC_PIV ) if LEFT=false.

This subroutine is adapted from:

Lawson, C.L., and Hanson, R.J., 1974:

Solving least square problems. Prentice-Hall.

`subroutine apply_h1 ( u, tau, vec )`¶

Purpose¶

APPLY_H1 applies a real elementary reflector H generated by H1 to a real vector C . H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

Arguments¶

U (INPUT) real(stnd), dimension(:)

On entry, U contains the vector v of the Householder reflector, as generated by H1.

TAU (INPUT) real(stnd)

On entry, the scalar tau of the Householder reflector, as generated by H1.

VEC (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the real vector C .

On exit, C is overwritten by the vector H * C .

Further Details¶

It is assumed that size( VEC ) >= size( U ) .

This subroutine is adapted from

Lawson, C.L., and Hanson, R.J., 1974: Solving least square problems.

Prentice-Hall.

`subroutine apply_h1 ( u, tau, mat, left )`¶

Purpose¶

APPLY_H1 applies a real elementary reflector H generated by H1 to a real n by m or m by n matrix, C, from the left or the right. H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

Arguments¶

U (INPUT) real(stnd), dimension(:)

On entry, U contains the vector v of the Householder reflector, as generated by H1.

TAU (INPUT) real(stnd)

On entry, the scalar tau of the Householder reflector, as generated by H1.

MAT (INPUT/OUTPUT) real(stnd), dimension(:,:)

On entry, the n by m or m by n matrix C.

On exit, C is overwritten by the matrix H * C (LEFT=true) or C * H (LEFT=false).

LEFT (INPUT) logical(lgl)

If:

LEFT=true, H is applied to the real matrix C from the left

LEFT=false, H is applied to the real matrix C from the right.

Further Details¶

It is assumed that:

size( MAT, 1 ) >= size( U ) if LEFT=true

size( MAT, 2 ) >= size( U ) if LEFT=false.

This subroutine is adapted from:

Lawson, C.L., and Hanson, R.J., 1974:

Solving least square problems. Prentice-Hall.

`subroutine apply_h2 ( u, up, tau, piv, vec )`¶

Purpose¶

APPLY_H2 applies a real elementary reflector H generated by H2 to a real vector C . H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

Arguments¶

U (INPUT) real(stnd), dimension(:)

On entry, U contains the vector v(2:) of the Householder reflector, as generated by H2.

UP (INPUT) real(stnd)

On entry, the value v(1) of the Householder reflector, as generated by H2.

TAU (INPUT) real(stnd)

On entry, the scalar tau of the Householder reflector, as generated by H2.

PIV (INPUT/OUTPUT) real(stnd)

On entry, the scalar C[1].

On exit, PIV is overwritten by the scalar (H * C)[1].

VEC (INPUT/OUTPUT) real(stnd), dimension(:)

On entry, the real vector C[2:].

On exit, C is overwritten by the vector (H * C)[2:].

Further Details¶

It is assumed that size( VEC ) >= size( U ) .

This subroutine is adapted from:

Lawson, C.L., and Hanson, R.J., 1974:

Solving least square problems. Prentice-Hall.

`subroutine apply_h2 ( u, up, tau, vec_piv, mat, left )`¶

Purpose¶

APPLY_H2 applies a real elementary reflector H generated by H2 to a real n by m or m by n matrix, C, from the left or the right. H is represented in the form

H = I + tau * ( v * v’ ) ,

where tau is a real scalar and v is a real n-element vector.

Arguments¶

U (INPUT) real(stnd), dimension(:)

On entry, U contains the vector v(2:) of the Householder reflector, as generated by H2.

UP (INPUT) real(stnd)

On entry, the value v(1) of the Householder reflector, as generated by H2.

TAU (INPUT) real(stnd)

On entry, the scalar tau of the Householder reflector, as generated by H2.

VEC_PIV (INPUT/OUTPUT) real(stnd), dimension(:)

If LEFT=true:

On entry, the row_vector C[1,:]

On exit, VEC_PIV is overwritten by the vector (H * C)[1,:].

If LEFT=false:

On entry, the column_vector C[:,1]

On exit, VEC_PIV is overwritten by the vector (C * H)[:,1].

MAT (INPUT/OUTPUT) real(stnd), dimension(:,:)

If LEFT=true:

On entry, the n-1 by m matrix C[2:,:]

On exit, MAT is overwritten by the matrix (H * C)[2:,:] .

If LEFT=false:

On entry, the m by n-1 matrix C[:,2:]

On exit, MAT is overwritten by the matrix (C * H)[:,2:].

LEFT (INPUT) logical(lgl)

If:

LEFT=true, H is applied to the real matrix C from the left

LEFT=false, H is applied to the real matrix C from the right.

Further Details¶

It is assumed that:

size( MAT, 1 ) >= size( U ) and size( MAT, 2 ) >= size( VEC_PIV ) if LEFT=true

size( MAT, 2 ) >= size( U ) and size( MAT, 1 ) >= size( VEC_PIV ) if LEFT=false.

This subroutine is adapted from:

Lawson, C.L., and Hanson, R.J., 1974:

Solving least square problems. Prentice-Hall.

Module_Hous_Procedures¶

subroutine hous1 ( u, tau )¶

Purpose¶

Arguments¶

Further Details¶

subroutine hous1 ( u, tau, beta )¶

Purpose¶

Arguments¶

Further Details¶

subroutine hous2 ( pivot, u, tau )¶

Purpose¶

Arguments¶

Further Details¶

subroutine apply_hous1 ( u, tau, vec )¶

Purpose¶

Arguments¶

Further Details¶

subroutine apply_hous1 ( u, tau, mat, left )¶

Purpose¶

Arguments¶

Further Details¶

subroutine apply_hous2 ( u, tau, piv, vec )¶

Purpose¶

Arguments¶

Further Details¶

subroutine apply_hous2 ( u, tau, vec_piv, mat, left )¶

Purpose¶

Arguments¶

Further Details¶

subroutine h1 ( u, beta, tau )¶

Purpose¶

Arguments¶

Further Details¶

subroutine h1 ( u, beta, tau, vec )¶

Purpose¶

Arguments¶

Further Details¶

subroutine h1 ( u, beta, tau, mat, left )¶

Purpose¶

Arguments¶

Further Details¶

subroutine h2 ( beta, u, up, tau )¶

Purpose¶

Arguments¶

Further Details¶

subroutine h2 ( beta, u, up, tau, piv, vec )¶

Purpose¶

Arguments¶

Further Details¶

subroutine h2 ( beta, u, up, tau, vec_piv, mat, left )¶

Purpose¶

Arguments¶

Further Details¶

subroutine apply_h1 ( u, tau, vec )¶

Purpose¶

Arguments¶

Further Details¶

subroutine apply_h1 ( u, tau, mat, left )¶

Purpose¶

Arguments¶

Further Details¶

subroutine apply_h2 ( u, up, tau, piv, vec )¶

Purpose¶

Arguments¶

Further Details¶

subroutine apply_h2 ( u, up, tau, vec_piv, mat, left )¶

Purpose¶

Arguments¶

Further Details¶

`subroutine hous1 ( u, tau )`¶

`subroutine hous1 ( u, tau, beta )`¶

`subroutine hous2 ( pivot, u, tau )`¶

`subroutine apply_hous1 ( u, tau, vec )`¶

`subroutine apply_hous1 ( u, tau, mat, left )`¶

`subroutine apply_hous2 ( u, tau, piv, vec )`¶

`subroutine apply_hous2 ( u, tau, vec_piv, mat, left )`¶

`subroutine h1 ( u, beta, tau )`¶

`subroutine h1 ( u, beta, tau, vec )`¶

`subroutine h1 ( u, beta, tau, mat, left )`¶

`subroutine h2 ( beta, u, up, tau )`¶

`subroutine h2 ( beta, u, up, tau, piv, vec )`¶

`subroutine h2 ( beta, u, up, tau, vec_piv, mat, left )`¶

`subroutine apply_h1 ( u, tau, vec )`¶

`subroutine apply_h1 ( u, tau, mat, left )`¶

`subroutine apply_h2 ( u, up, tau, piv, vec )`¶

`subroutine apply_h2 ( u, up, tau, vec_piv, mat, left )`¶