# 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.

## 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:**

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

## Author

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