---
name: "ssytrd.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/ssytrd.f"
---

# ssytrd.f(3)

## NAME

SRC/ssytrd.f

## SYNOPSIS

### Functions/Subroutines

subroutine **ssytrd** (uplo, n, a, lda, d, e, tau, work, lwork, info)\
 **SSYTRD**

## Function/Subroutine Documentation

### subroutine ssytrd (character uplo, integer n, real, dimension( lda, \* ) a, integer lda, real, dimension( \* ) d, real, dimension( \* ) e, real, dimension( \* ) tau, real, dimension( \* ) work, integer lwork, integer info)

**SSYTRD**

**Purpose:**

```
!>
!> SSYTRD reduces a real symmetric matrix A to real symmetric
!> tridiagonal form T by an orthogonal similarity transformation:
!> Q**T * A * Q = T.
!>
```

**Parameters**

*UPLO*

```
!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored.
!>
```

*N*

```
!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>
```

*A*

```
!>          A is REAL array, dimension (LDA,N)
!>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
!>          N-by-N upper triangular part of A contains the upper
!>          triangular part of the matrix A, and the strictly lower
!>          triangular part of A is not referenced.  If UPLO = 'L', the
!>          leading N-by-N lower triangular part of A contains the lower
!>          triangular part of the matrix A, and the strictly upper
!>          triangular part of A is not referenced.
!>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
!>          of A are overwritten by the corresponding elements of the
!>          tridiagonal matrix T, and the elements above the first
!>          superdiagonal, with the array TAU, represent the orthogonal
!>          matrix Q as a product of elementary reflectors; if UPLO
!>          = 'L', the diagonal and first subdiagonal of A are over-
!>          written by the corresponding elements of the tridiagonal
!>          matrix T, and the elements below the first subdiagonal, with
!>          the array TAU, represent the orthogonal matrix Q as a product
!>          of elementary reflectors. See Further Details.
!>
```

*LDA*

```
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
```

*D*

```
!>          D is REAL array, dimension (N)
!>          The diagonal elements of the tridiagonal matrix T:
!>          D(i) = A(i,i).
!>
```

*E*

```
!>          E is REAL array, dimension (N-1)
!>          The off-diagonal elements of the tridiagonal matrix T:
!>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
!>
```

*TAU*

```
!>          TAU is REAL array, dimension (N-1)
!>          The scalar factors of the elementary reflectors (see Further
!>          Details).
!>
```

*WORK*

```
!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
```

*LWORK*

```
!>          LWORK is INTEGER
!>          The dimension of the array WORK.  LWORK >= 1.
!>          For optimum performance LWORK >= N*NB, where NB is the
!>          optimal blocksize.
!>
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the WORK array, returns
!>          this value as the first entry of the WORK array, and no error
!>          message related to LWORK is issued by XERBLA.
!>
```

*INFO*

```
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

```
!>
!>  If UPLO = 'U', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(n-1) . . . H(2) H(1).
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**T
!>
!>  where tau is a real scalar, and v is a real vector with
!>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
!>  A(1:i-1,i+1), and tau in TAU(i).
!>
!>  If UPLO = 'L', the matrix Q is represented as a product of elementary
!>  reflectors
!>
!>     Q = H(1) H(2) . . . H(n-1).
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - tau * v * v**T
!>
!>  where tau is a real scalar, and v is a real vector with
!>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
!>  and tau in TAU(i).
!>
!>  The contents of A on exit are illustrated by the following examples
!>  with n = 5:
!>
!>  if UPLO = 'U':                       if UPLO = 'L':
!>
!>    (  d   e   v2  v3  v4 )              (  d                  )
!>    (      d   e   v3  v4 )              (  e   d              )
!>    (          d   e   v4 )              (  v1  e   d          )
!>    (              d   e  )              (  v1  v2  e   d      )
!>    (                  d  )              (  v1  v2  v3  e   d  )
!>
!>  where d and e denote diagonal and off-diagonal elements of T, and vi
!>  denotes an element of the vector defining H(i).
!>
```

Definition at line **191** of file **ssytrd.f**.

## Author

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