# laev2 - Man Page

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.

## 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

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.```

Definition at line 120 of file claev2.f.

### 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

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.```

Definition at line 119 of file dlaev2.f.

### 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

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.```

Definition at line 119 of file slaev2.f.

### 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

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.```

Definition at line 120 of file zlaev2.f.

## Author

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## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK