---
name: "dsyt21.f"
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/dsyt21.f"
---

# dsyt21.f(3)

## NAME

TESTING/EIG/dsyt21.f

## SYNOPSIS

### Functions/Subroutines

subroutine **dsyt21** (itype, uplo, n, kband, a, lda, d, e, u, ldu, v, ldv, tau, work, result)\
 **DSYT21**

## Function/Subroutine Documentation

### subroutine dsyt21 (integer itype, character uplo, integer n, integer kband, double precision, dimension( lda, \* ) a, integer lda, double precision, dimension( \* ) d, double precision, dimension( \* ) e, double precision, dimension( ldu, \* ) u, integer ldu, double precision, dimension( ldv, \* ) v, integer ldv, double precision, dimension( \* ) tau, double precision, dimension( \* ) work, double precision, dimension( 2 ) result)

**DSYT21**

**Purpose:**

```
!>
!> DSYT21 generally checks a decomposition of the form
!>
!>    A = U S U**T
!>
!> where **T means transpose, A is symmetric, U is orthogonal, and S is
!> diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1).
!>
!> If ITYPE=1, then U is represented as a dense matrix; otherwise U is
!> expressed as a product of Householder transformations, whose vectors
!> are stored in the array  and whose scaling constants are in .
!> We shall use the letter  to refer to the product of Householder
!> transformations (which should be equal to U).
!>
!> Specifically, if ITYPE=1, then:
!>
!>    RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and
!>    RESULT(2) = | I - U U**T | / ( n ulp )
!>
!> If ITYPE=2, then:
!>
!>    RESULT(1) = | A - V S V**T | / ( |A| n ulp )
!>
!> If ITYPE=3, then:
!>
!>    RESULT(1) = | I - V U**T | / ( n ulp )
!>
!> For ITYPE > 1, the transformation U is expressed as a product
!> V = H(1)...H(n-2),  where H(j) = I  -  tau(j) v(j) v(j)**T and each
!> vector v(j) has its first j elements 0 and the remaining n-j elements
!> stored in V(j+1:n,j).
!>
```

**Parameters**

*ITYPE*

```
!>          ITYPE is INTEGER
!>          Specifies the type of tests to be performed.
!>          1: U expressed as a dense orthogonal matrix:
!>             RESULT(1) = | A - U S U**T | / ( |A| n ulp )  and
!>             RESULT(2) = | I - U U**T | / ( n ulp )
!>
!>          2: U expressed as a product V of Housholder transformations:
!>             RESULT(1) = | A - V S V**T | / ( |A| n ulp )
!>
!>          3: U expressed both as a dense orthogonal matrix and
!>             as a product of Housholder transformations:
!>             RESULT(1) = | I - V U**T | / ( n ulp )
!>
```

*UPLO*

```
!>          UPLO is CHARACTER
!>          If UPLO='U', the upper triangle of A and V will be used and
!>          the (strictly) lower triangle will not be referenced.
!>          If UPLO='L', the lower triangle of A and V will be used and
!>          the (strictly) upper triangle will not be referenced.
!>
```

*N*

```
!>          N is INTEGER
!>          The size of the matrix.  If it is zero, DSYT21 does nothing.
!>          It must be at least zero.
!>
```

*KBAND*

```
!>          KBAND is INTEGER
!>          The bandwidth of the matrix.  It may only be zero or one.
!>          If zero, then S is diagonal, and E is not referenced.  If
!>          one, then S is symmetric tri-diagonal.
!>
```

*A*

```
!>          A is DOUBLE PRECISION array, dimension (LDA, N)
!>          The original (unfactored) matrix.  It is assumed to be
!>          symmetric, and only the upper (UPLO='U') or only the lower
!>          (UPLO='L') will be referenced.
!>
```

*LDA*

```
!>          LDA is INTEGER
!>          The leading dimension of A.  It must be at least 1
!>          and at least N.
!>
```

*D*

```
!>          D is DOUBLE PRECISION array, dimension (N)
!>          The diagonal of the (symmetric tri-) diagonal matrix.
!>
```

*E*

```
!>          E is DOUBLE PRECISION array, dimension (N-1)
!>          The off-diagonal of the (symmetric tri-) diagonal matrix.
!>          E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
!>          (3,2) element, etc.
!>          Not referenced if KBAND=0.
!>
```

*U*

```
!>          U is DOUBLE PRECISION array, dimension (LDU, N)
!>          If ITYPE=1 or 3, this contains the orthogonal matrix in
!>          the decomposition, expressed as a dense matrix.  If ITYPE=2,
!>          then it is not referenced.
!>
```

*LDU*

```
!>          LDU is INTEGER
!>          The leading dimension of U.  LDU must be at least N and
!>          at least 1.
!>
```

*V*

```
!>          V is DOUBLE PRECISION array, dimension (LDV, N)
!>          If ITYPE=2 or 3, the columns of this array contain the
!>          Householder vectors used to describe the orthogonal matrix
!>          in the decomposition.  If UPLO='L', then the vectors are in
!>          the lower triangle, if UPLO='U', then in the upper
!>          triangle.
!>          *NOTE* If ITYPE=2 or 3, V is modified and restored.  The
!>          subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
!>          is set to one, and later reset to its original value, during
!>          the course of the calculation.
!>          If ITYPE=1, then it is neither referenced nor modified.
!>
```

*LDV*

```
!>          LDV is INTEGER
!>          The leading dimension of V.  LDV must be at least N and
!>          at least 1.
!>
```

*TAU*

```
!>          TAU is DOUBLE PRECISION array, dimension (N)
!>          If ITYPE >= 2, then TAU(j) is the scalar factor of
!>          v(j) v(j)**T in the Householder transformation H(j) of
!>          the product  U = H(1)...H(n-2)
!>          If ITYPE < 2, then TAU is not referenced.
!>
```

*WORK*

```
!>          WORK is DOUBLE PRECISION array, dimension (2*N**2)
!>
```

*RESULT*

```
!>          RESULT is DOUBLE PRECISION array, dimension (2)
!>          The values computed by the two tests described above.  The
!>          values are currently limited to 1/ulp, to avoid overflow.
!>          RESULT(1) is always modified.  RESULT(2) is modified only
!>          if ITYPE=1.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **205** of file **dsyt21.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.