---
name: "laev2"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/laev2"
---

# laev2(3)

laev2: 2x2 eig

## 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.
!>
```

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

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.
!>
```

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

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.
!>
```

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

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.
!>
```

Definition at line **120** of file **zlaev2.f**.

## Author

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