---
name: "dlahqr.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/dlahqr.f"
---

# dlahqr.f(3)

## NAME

SRC/dlahqr.f

## SYNOPSIS

### Functions/Subroutines

subroutine **dlahqr** (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info)\
 **DLAHQR** computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

## Function/Subroutine Documentation

### subroutine dlahqr (logical wantt, logical wantz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, \* ) h, integer ldh, double precision, dimension( \* ) wr, double precision, dimension( \* ) wi, integer iloz, integer ihiz, double precision, dimension( ldz, \* ) z, integer ldz, integer info)

**DLAHQR** computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

**Purpose:**

```
!>
!>    DLAHQR is an auxiliary routine called by DHSEQR to update the
!>    eigenvalues and Schur decomposition already computed by DHSEQR, by
!>    dealing with the Hessenberg submatrix in rows and columns ILO to
!>    IHI.
!>
```

**Parameters**

*WANTT*

```
!>          WANTT is LOGICAL
!>          = .TRUE. : the full Schur form T is required;
!>          = .FALSE.: only eigenvalues are required.
!>
```

*WANTZ*

```
!>          WANTZ is LOGICAL
!>          = .TRUE. : the matrix of Schur vectors Z is required;
!>          = .FALSE.: Schur vectors are not required.
!>
```

*N*

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

*ILO*

```
!>          ILO is INTEGER
!>
```

*IHI*

```
!>          IHI is INTEGER
!>          It is assumed that H is already upper quasi-triangular in
!>          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
!>          ILO = 1). DLAHQR works primarily with the Hessenberg
!>          submatrix in rows and columns ILO to IHI, but applies
!>          transformations to all of H if WANTT is .TRUE..
!>          1 <= ILO <= max(1,IHI); IHI <= N.
!>
```

*H*

```
!>          H is DOUBLE PRECISION array, dimension (LDH,N)
!>          On entry, the upper Hessenberg matrix H.
!>          On exit, if INFO is zero and if WANTT is .TRUE., H is upper
!>          quasi-triangular in rows and columns ILO:IHI, with any
!>          2-by-2 diagonal blocks in standard form. If INFO is zero
!>          and WANTT is .FALSE., the contents of H are unspecified on
!>          exit.  The output state of H if INFO is nonzero is given
!>          below under the description of INFO.
!>
```

*LDH*

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

*WR*

```
!>          WR is DOUBLE PRECISION array, dimension (N)
!>
```

*WI*

```
!>          WI is DOUBLE PRECISION array, dimension (N)
!>          The real and imaginary parts, respectively, of the computed
!>          eigenvalues ILO to IHI are stored in the corresponding
!>          elements of WR and WI. If two eigenvalues are computed as a
!>          complex conjugate pair, they are stored in consecutive
!>          elements of WR and WI, say the i-th and (i+1)th, with
!>          WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
!>          eigenvalues are stored in the same order as on the diagonal
!>          of the Schur form returned in H, with WR(i) = H(i,i), and, if
!>          H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
!>          WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).
!>
```

*ILOZ*

```
!>          ILOZ is INTEGER
!>
```

*IHIZ*

```
!>          IHIZ is INTEGER
!>          Specify the rows of Z to which transformations must be
!>          applied if WANTZ is .TRUE..
!>          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
!>
```

*Z*

```
!>          Z is DOUBLE PRECISION array, dimension (LDZ,N)
!>          If WANTZ is .TRUE., on entry Z must contain the current
!>          matrix Z of transformations accumulated by DHSEQR, and on
!>          exit Z has been updated; transformations are applied only to
!>          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
!>          If WANTZ is .FALSE., Z is not referenced.
!>
```

*LDZ*

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

*INFO*

```
!>          INFO is INTEGER
!>           = 0:  successful exit
!>           > 0:  If INFO = i, DLAHQR failed to compute all the
!>                  eigenvalues ILO to IHI in a total of 30 iterations
!>                  per eigenvalue; elements i+1:ihi of WR and WI
!>                  contain those eigenvalues which have been
!>                  successfully computed.
!>
!>                  If INFO > 0 and WANTT is .FALSE., then on exit,
!>                  the remaining unconverged eigenvalues are the
!>                  eigenvalues of the upper Hessenberg matrix rows
!>                  and columns ILO through INFO of the final, output
!>                  value of H.
!>
!>                  If INFO > 0 and WANTT is .TRUE., then on exit
!>          (*)       (initial value of H)*U  = U*(final value of H)
!>                  where U is an orthogonal matrix.    The final
!>                  value of H is upper Hessenberg and triangular in
!>                  rows and columns INFO+1 through IHI.
!>
!>                  If INFO > 0 and WANTZ is .TRUE., then on exit
!>                      (final value of Z)  = (initial value of Z)*U
!>                  where U is the orthogonal matrix in (*)
!>                  (regardless of the value of WANTT.)
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

```
!>
!>     02-96 Based on modifications by
!>     David Day, Sandia National Laboratory, USA
!>
!>     12-04 Further modifications by
!>     Ralph Byers, University of Kansas, USA
!>     This is a modified version of DLAHQR from LAPACK version 3.0.
!>     It is (1) more robust against overflow and underflow and
!>     (2) adopts the more conservative Ahues & Tisseur stopping
!>     criterion (LAWN 122, 1997).
!>
```

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

## Author

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