laev2(3)

NAME
laev2 - laev2: 2x2 eig

SYNOPSIS

  Functions

subroutine claev2 (a, b, c, rt1, rt2, cs1, sn1)
CLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
subroutine dlaev2 (a, b, c, rt1, rt2, cs1, sn1)
DLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
subroutine slaev2 (a, b, c, rt1, rt2, cs1, sn1)
SLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
subroutine zlaev2 (a, b, c, rt1, rt2, cs1, sn1)
ZLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

Detailed Description

Function Documentation

  subroutine claev2 (complex a, complex b, complex c, real rt1, real rt2, real cs1, complex sn1)

CLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

Purpose:

    !>
    !> CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
    !>    [  A         B  ]
    !>    [  CONJG(B)  C  ].
    !> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
    !> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
    !> eigenvector for RT1, giving the decomposition
    !>
    !> [ CS1  CONJG(SN1) ] [    A     B ] [ CS1 -CONJG(SN1) ] = [ RT1  0  ]
    !> [-SN1     CS1     ] [ CONJG(B) C ] [ SN1     CS1     ]   [  0  RT2 ].
    !>

Parameters
    A

    !>          A is COMPLEX
    !>         The (1,1) element of the 2-by-2 matrix.
    !>

    B

    !>          B is COMPLEX
    !>         The (1,2) element and the conjugate of the (2,1) element of
    !>         the 2-by-2 matrix.
    !>

    C

    !>          C is COMPLEX
    !>         The (2,2) element of the 2-by-2 matrix.
    !>

    RT1

    !>          RT1 is REAL
    !>         The eigenvalue of larger absolute value.
    !>

    RT2

    !>          RT2 is REAL
    !>         The eigenvalue of smaller absolute value.
    !>

    CS1

    !>          CS1 is REAL
    !>

    SN1

    !>          SN1 is COMPLEX
    !>         The vector (CS1, SN1) is a unit right eigenvector for RT1.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  RT1 is accurate to a few ulps barring over/underflow.
    !>
    !>  RT2 may be inaccurate if there is massive cancellation in the
    !>  determinant A*C-B*B; higher precision or correctly rounded or
    !>  correctly truncated arithmetic would be needed to compute RT2
    !>  accurately in all cases.
    !>
    !>  CS1 and SN1 are accurate to a few ulps barring over/underflow.
    !>
    !>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
    !>  Underflow is harmless if the input data is 0 or exceeds
    !>     underflow_threshold / macheps.
    !>

  subroutine dlaev2 (double precision a, double precision b, double precision c, double precision rt1, double precision rt2, double precision cs1, double precision sn1)

DLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

Purpose:

    !>
    !> DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
    !>    [  A   B  ]
    !>    [  B   C  ].
    !> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
    !> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
    !> eigenvector for RT1, giving the decomposition
    !>
    !>    [ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
    !>    [-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].
    !>

Parameters
    A

    !>          A is DOUBLE PRECISION
    !>          The (1,1) element of the 2-by-2 matrix.
    !>

    B

    !>          B is DOUBLE PRECISION
    !>          The (1,2) element and the conjugate of the (2,1) element of
    !>          the 2-by-2 matrix.
    !>

    C

    !>          C is DOUBLE PRECISION
    !>          The (2,2) element of the 2-by-2 matrix.
    !>

    RT1

    !>          RT1 is DOUBLE PRECISION
    !>          The eigenvalue of larger absolute value.
    !>

    RT2

    !>          RT2 is DOUBLE PRECISION
    !>          The eigenvalue of smaller absolute value.
    !>

    CS1

    !>          CS1 is DOUBLE PRECISION
    !>

    SN1

    !>          SN1 is DOUBLE PRECISION
    !>          The vector (CS1, SN1) is a unit right eigenvector for RT1.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  RT1 is accurate to a few ulps barring over/underflow.
    !>
    !>  RT2 may be inaccurate if there is massive cancellation in the
    !>  determinant A*C-B*B; higher precision or correctly rounded or
    !>  correctly truncated arithmetic would be needed to compute RT2
    !>  accurately in all cases.
    !>
    !>  CS1 and SN1 are accurate to a few ulps barring over/underflow.
    !>
    !>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
    !>  Underflow is harmless if the input data is 0 or exceeds
    !>     underflow_threshold / macheps.
    !>

  subroutine slaev2 (real a, real b, real c, real rt1, real rt2, real cs1, real sn1)

SLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

Purpose:

    !>
    !> SLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
    !>    [  A   B  ]
    !>    [  B   C  ].
    !> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
    !> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
    !> eigenvector for RT1, giving the decomposition
    !>
    !>    [ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
    !>    [-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].
    !>

Parameters
    A

    !>          A is REAL
    !>          The (1,1) element of the 2-by-2 matrix.
    !>

    B

    !>          B is REAL
    !>          The (1,2) element and the conjugate of the (2,1) element of
    !>          the 2-by-2 matrix.
    !>

    C

    !>          C is REAL
    !>          The (2,2) element of the 2-by-2 matrix.
    !>

    RT1

    !>          RT1 is REAL
    !>          The eigenvalue of larger absolute value.
    !>

    RT2

    !>          RT2 is REAL
    !>          The eigenvalue of smaller absolute value.
    !>

    CS1

    !>          CS1 is REAL
    !>

    SN1

    !>          SN1 is REAL
    !>          The vector (CS1, SN1) is a unit right eigenvector for RT1.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  RT1 is accurate to a few ulps barring over/underflow.
    !>
    !>  RT2 may be inaccurate if there is massive cancellation in the
    !>  determinant A*C-B*B; higher precision or correctly rounded or
    !>  correctly truncated arithmetic would be needed to compute RT2
    !>  accurately in all cases.
    !>
    !>  CS1 and SN1 are accurate to a few ulps barring over/underflow.
    !>
    !>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
    !>  Underflow is harmless if the input data is 0 or exceeds
    !>     underflow_threshold / macheps.
    !>

  subroutine zlaev2 (complex*16 a, complex*16 b, complex*16 c, double precision rt1, double precision rt2, double precision cs1, complex*16 sn1)

ZLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

Purpose:

    !>
    !> ZLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
    !>    [  A         B  ]
    !>    [  CONJG(B)  C  ].
    !> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
    !> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
    !> eigenvector for RT1, giving the decomposition
    !>
    !> [ CS1  CONJG(SN1) ] [    A     B ] [ CS1 -CONJG(SN1) ] = [ RT1  0  ]
    !> [-SN1     CS1     ] [ CONJG(B) C ] [ SN1     CS1     ]   [  0  RT2 ].
    !>

Parameters
    A

    !>          A is COMPLEX*16
    !>         The (1,1) element of the 2-by-2 matrix.
    !>

    B

    !>          B is COMPLEX*16
    !>         The (1,2) element and the conjugate of the (2,1) element of
    !>         the 2-by-2 matrix.
    !>

    C

    !>          C is COMPLEX*16
    !>         The (2,2) element of the 2-by-2 matrix.
    !>

    RT1

    !>          RT1 is DOUBLE PRECISION
    !>         The eigenvalue of larger absolute value.
    !>

    RT2

    !>          RT2 is DOUBLE PRECISION
    !>         The eigenvalue of smaller absolute value.
    !>

    CS1

    !>          CS1 is DOUBLE PRECISION
    !>

    SN1

    !>          SN1 is COMPLEX*16
    !>         The vector (CS1, SN1) is a unit right eigenvector for RT1.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  RT1 is accurate to a few ulps barring over/underflow.
    !>
    !>  RT2 may be inaccurate if there is massive cancellation in the
    !>  determinant A*C-B*B; higher precision or correctly rounded or
    !>  correctly truncated arithmetic would be needed to compute RT2
    !>  accurately in all cases.
    !>
    !>  CS1 and SN1 are accurate to a few ulps barring over/underflow.
    !>
    !>  Overflow is possible only if RT1 is within a factor of 5 of overflow.
    !>  Underflow is harmless if the input data is 0 or exceeds
    !>     underflow_threshold / macheps.
    !>

Author

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