## table of contents

realSYcomputational(3) | LAPACK | realSYcomputational(3) |

# NAME¶

realSYcomputational - real

# SYNOPSIS¶

## Functions¶

subroutine **sla_syamv** (UPLO, N, ALPHA, A, LDA, X, INCX,
BETA, Y, INCY)

**SLA_SYAMV** computes a matrix-vector product using a symmetric indefinite
matrix to calculate error bounds. real function **sla_syrcond** (UPLO, N,
A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)

**SLA_SYRCOND** estimates the Skeel condition number for a symmetric
indefinite matrix. subroutine **sla_syrfsx_extended** (PREC_TYPE, UPLO,
N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT,
N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH,
RTHRESH, DZ_UB, IGNORE_CWISE, INFO)

**SLA_SYRFSX_EXTENDED** improves the computed solution to a system of
linear equations for symmetric indefinite matrices by performing
extra-precise iterative refinement and provides error bounds and backward
error estimates for the solution. real function **sla_syrpvgrw** (UPLO,
N, INFO, A, LDA, AF, LDAF, IPIV, WORK)

**SLA_SYRPVGRW** computes the reciprocal pivot growth factor
norm(A)/norm(U) for a symmetric indefinite matrix. subroutine **slasyf**
(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)

**SLASYF** computes a partial factorization of a real symmetric matrix
using the Bunch-Kaufman diagonal pivoting method. subroutine
**slasyf_aa** (UPLO, J1, M, NB, A, LDA, IPIV, H, LDH, WORK)

**SLASYF_AA** subroutine **slasyf_rook** (UPLO, N, NB, KB, A, LDA, IPIV,
W, LDW, INFO)

**SLASYF_ROOK** computes a partial factorization of a real symmetric matrix
using the bounded Bunch-Kaufman ('rook') diagonal pivoting method.
subroutine **ssycon** (UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK,
INFO)

**SSYCON** subroutine **ssycon_rook** (UPLO, N, A, LDA, IPIV, ANORM,
RCOND, WORK, IWORK, INFO)

** SSYCON_ROOK ** subroutine **ssyconv** (UPLO, WAY, N, A, LDA, IPIV, E,
INFO)

**SSYCONV** subroutine **ssyequb** (UPLO, N, A, LDA, S, SCOND, AMAX,
WORK, INFO)

**SSYEQUB** subroutine **ssygs2** (ITYPE, UPLO, N, A, LDA, B, LDB, INFO)

**SSYGS2** reduces a symmetric definite generalized eigenproblem to
standard form, using the factorization results obtained from spotrf
(unblocked algorithm). subroutine **ssygst** (ITYPE, UPLO, N, A, LDA, B,
LDB, INFO)

**SSYGST** subroutine **ssyrfs** (UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)

**SSYRFS** subroutine **ssyrfsx** (UPLO, EQUED, N, NRHS, A, LDA, AF,
LDAF, IPIV, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)

**SSYRFSX** subroutine **ssytd2** (UPLO, N, A, LDA, D, E, TAU, INFO)

**SSYTD2** reduces a symmetric matrix to real symmetric tridiagonal form by
an orthogonal similarity transformation (unblocked algorithm). subroutine
**ssytf2** (UPLO, N, A, LDA, IPIV, INFO)

**SSYTF2** computes the factorization of a real symmetric indefinite
matrix, using the diagonal pivoting method (unblocked algorithm). subroutine
**ssytf2_rook** (UPLO, N, A, LDA, IPIV, INFO)

**SSYTF2_ROOK** computes the factorization of a real symmetric indefinite
matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting method
(unblocked algorithm). subroutine **ssytrd** (UPLO, N, A, LDA, D, E, TAU,
WORK, LWORK, INFO)

**SSYTRD** subroutine **ssytrd_2stage** (VECT, UPLO, N, A, LDA, D, E,
TAU, HOUS2, LHOUS2, WORK, LWORK, INFO)

**SSYTRD_2STAGE** subroutine **ssytrd_sy2sb** (UPLO, N, KD, A, LDA, AB,
LDAB, TAU, WORK, LWORK, INFO)

**SSYTRD_SY2SB** subroutine **ssytrf** (UPLO, N, A, LDA, IPIV, WORK,
LWORK, INFO)

**SSYTRF** subroutine **ssytrf_aa** (UPLO, N, A, LDA, IPIV, WORK, LWORK,
INFO)

**SSYTRF_AA** subroutine **ssytrf_aa_2stage** (UPLO, N, A, LDA, TB, LTB,
IPIV, IPIV2, WORK, LWORK, INFO)

**SSYTRF_AA_2STAGE** subroutine **ssytrf_rook** (UPLO, N, A, LDA, IPIV,
WORK, LWORK, INFO)

**SSYTRF_ROOK** subroutine **ssytri** (UPLO, N, A, LDA, IPIV, WORK,
INFO)

**SSYTRI** subroutine **ssytri2** (UPLO, N, A, LDA, IPIV, WORK, LWORK,
INFO)

**SSYTRI2** subroutine **ssytri2x** (UPLO, N, A, LDA, IPIV, WORK, NB,
INFO)

**SSYTRI2X** subroutine **ssytri_rook** (UPLO, N, A, LDA, IPIV, WORK,
INFO)

**SSYTRI_ROOK** subroutine **ssytrs** (UPLO, N, NRHS, A, LDA, IPIV, B,
LDB, INFO)

**SSYTRS** subroutine **ssytrs2** (UPLO, N, NRHS, A, LDA, IPIV, B, LDB,
WORK, INFO)

**SSYTRS2** subroutine **ssytrs_aa** (UPLO, N, NRHS, A, LDA, IPIV, B,
LDB, WORK, LWORK, INFO)

**SSYTRS_AA** subroutine **ssytrs_aa_2stage** (UPLO, N, NRHS, A, LDA,
TB, LTB, IPIV, IPIV2, B, LDB, INFO)

**SSYTRS_AA_2STAGE** subroutine **ssytrs_rook** (UPLO, N, NRHS, A, LDA,
IPIV, B, LDB, INFO)

**SSYTRS_ROOK** subroutine **stgsyl** (TRANS, IJOB, M, N, A, LDA, B,
LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)

**STGSYL** subroutine **strsyl** (TRANA, TRANB, ISGN, M, N, A, LDA, B,
LDB, C, LDC, SCALE, INFO)

**STRSYL**

# Detailed Description¶

This is the group of real computational functions for SY matrices

# Function Documentation¶

## subroutine sla_syamv (integer UPLO, integer N, real ALPHA, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) X, integer INCX, real BETA, real, dimension( * ) Y, integer INCY)¶

**SLA_SYAMV** computes a matrix-vector product using a
symmetric indefinite matrix to calculate error bounds.

**Purpose:**

SLA_SYAMV performs the matrix-vector operation

y := alpha*abs(A)*abs(x) + beta*abs(y),

where alpha and beta are scalars, x and y are vectors and A is an

n by n symmetric matrix.

This function is primarily used in calculating error bounds.

To protect against underflow during evaluation, components in

the resulting vector are perturbed away from zero by (N+1)

times the underflow threshold. To prevent unnecessarily large

errors for block-structure embedded in general matrices,

"symbolically" zero components are not perturbed. A zero

entry is considered "symbolic" if all multiplications involved

in computing that entry have at least one zero multiplicand.

**Parameters**

*UPLO*

UPLO is INTEGER

On entry, UPLO specifies whether the upper or lower

triangular part of the array A is to be referenced as

follows:

UPLO = BLAS_UPPER Only the upper triangular part of A

is to be referenced.

UPLO = BLAS_LOWER Only the lower triangular part of A

is to be referenced.

Unchanged on exit.

*N*

N is INTEGER

On entry, N specifies the number of columns of the matrix A.

N must be at least zero.

Unchanged on exit.

*ALPHA*

ALPHA is REAL .

On entry, ALPHA specifies the scalar alpha.

Unchanged on exit.

*A*

A is REAL array, dimension ( LDA, n ).

Before entry, the leading m by n part of the array A must

contain the matrix of coefficients.

Unchanged on exit.

*LDA*

LDA is INTEGER

On entry, LDA specifies the first dimension of A as declared

in the calling (sub) program. LDA must be at least

max( 1, n ).

Unchanged on exit.

*X*

X is REAL array, dimension

( 1 + ( n - 1 )*abs( INCX ) )

Before entry, the incremented array X must contain the

vector x.

Unchanged on exit.

*INCX*

INCX is INTEGER

On entry, INCX specifies the increment for the elements of

X. INCX must not be zero.

Unchanged on exit.

*BETA*

BETA is REAL .

On entry, BETA specifies the scalar beta. When BETA is

supplied as zero then Y need not be set on input.

Unchanged on exit.

*Y*

Y is REAL array, dimension

( 1 + ( n - 1 )*abs( INCY ) )

Before entry with BETA non-zero, the incremented array Y

must contain the vector y. On exit, Y is overwritten by the

updated vector y.

*INCY*

INCY is INTEGER

On entry, INCY specifies the increment for the elements of

Y. INCY must not be zero.

Unchanged on exit.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Level 2 Blas routine.

-- Written on 22-October-1986.

Jack Dongarra, Argonne National Lab.

Jeremy Du Croz, Nag Central Office.

Sven Hammarling, Nag Central Office.

Richard Hanson, Sandia National Labs.

-- Modified for the absolute-value product, April 2006

Jason Riedy, UC Berkeley

## real function sla_syrcond (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, integer CMODE, real, dimension( * ) C, integer INFO, real, dimension( * ) WORK, integer, dimension( * ) IWORK)¶

**SLA_SYRCOND** estimates the Skeel condition number for a
symmetric indefinite matrix.

**Purpose:**

SLA_SYRCOND estimates the Skeel condition number of op(A) * op2(C)

where op2 is determined by CMODE as follows

CMODE = 1 op2(C) = C

CMODE = 0 op2(C) = I

CMODE = -1 op2(C) = inv(C)

The Skeel condition number cond(A) = norminf( |inv(A)||A| )

is computed by computing scaling factors R such that

diag(R)*A*op2(C) is row equilibrated and computing the standard

infinity-norm condition number.

**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 number of linear equations, i.e., the order of the

matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER

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

*AF*

AF is REAL array, dimension (LDAF,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*CMODE*

CMODE is INTEGER

Determines op2(C) in the formula op(A) * op2(C) as follows:

CMODE = 1 op2(C) = C

CMODE = 0 op2(C) = I

CMODE = -1 op2(C) = inv(C)

*C*

C is REAL array, dimension (N)

The vector C in the formula op(A) * op2(C).

*INFO*

INFO is INTEGER

= 0: Successful exit.

i > 0: The ith argument is invalid.

*WORK*

WORK is REAL array, dimension (3*N).

Workspace.

*IWORK*

IWORK is INTEGER array, dimension (N).

Workspace.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine sla_syrfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldy, * ) Y, integer LDY, real, dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, real, dimension( * ) RES, real, dimension( * ) AYB, real, dimension( * ) DY, real, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)¶

**SLA_SYRFSX_EXTENDED** improves the computed solution to a
system of linear equations for symmetric indefinite matrices by performing
extra-precise iterative refinement and provides error bounds and backward
error estimates for the solution.

**Purpose:**

SLA_SYRFSX_EXTENDED improves the computed solution to a system of

linear equations by performing extra-precise iterative refinement

and provides error bounds and backward error estimates for the solution.

This subroutine is called by SSYRFSX to perform iterative refinement.

In addition to normwise error bound, the code provides maximum

componentwise error bound if possible. See comments for ERR_BNDS_NORM

and ERR_BNDS_COMP for details of the error bounds. Note that this

subroutine is only resonsible for setting the second fields of

ERR_BNDS_NORM and ERR_BNDS_COMP.

**Parameters**

*PREC_TYPE*

PREC_TYPE is INTEGER

Specifies the intermediate precision to be used in refinement.

The value is defined by ILAPREC(P) where P is a CHARACTER and P

= 'S': Single

= 'D': Double

= 'I': Indigenous

= 'X' or 'E': Extra

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*N*

N is INTEGER

The number of linear equations, i.e., the order of the

matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right-hand-sides, i.e., the number of columns of the

matrix B.

*A*

A is REAL array, dimension (LDA,N)

On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER

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

*AF*

AF is REAL array, dimension (LDAF,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*COLEQU*

COLEQU is LOGICAL

If .TRUE. then column equilibration was done to A before calling

this routine. This is needed to compute the solution and error

bounds correctly.

*C*

C is REAL array, dimension (N)

The column scale factors for A. If COLEQU = .FALSE., C

is not accessed. If C is input, each element of C should be a power

of the radix to ensure a reliable solution and error estimates.

Scaling by powers of the radix does not cause rounding errors unless

the result underflows or overflows. Rounding errors during scaling

lead to refining with a matrix that is not equivalent to the

input matrix, producing error estimates that may not be

reliable.

*B*

B is REAL array, dimension (LDB,NRHS)

The right-hand-side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*Y*

Y is REAL array, dimension (LDY,NRHS)

On entry, the solution matrix X, as computed by SSYTRS.

On exit, the improved solution matrix Y.

*LDY*

LDY is INTEGER

The leading dimension of the array Y. LDY >= max(1,N).

*BERR_OUT*

BERR_OUT is REAL array, dimension (NRHS)

On exit, BERR_OUT(j) contains the componentwise relative backward

error for right-hand-side j from the formula

max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )

where abs(Z) is the componentwise absolute value of the matrix

or vector Z. This is computed by SLA_LIN_BERR.

*N_NORMS*

N_NORMS is INTEGER

Determines which error bounds to return (see ERR_BNDS_NORM

and ERR_BNDS_COMP).

If N_NORMS >= 1 return normwise error bounds.

If N_NORMS >= 2 return componentwise error bounds.

*ERR_BNDS_NORM*

ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

max_j (abs(XTRUE(j,i) - X(j,i)))

------------------------------

max_j abs(X(j,i))

The array is indexed by the type of error information as described

below. There currently are up to three pieces of information

returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith

right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following

three fields:

err = 1 "Trust/don't trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * slamch('Epsilon').

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * slamch('Epsilon'). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated normwise

reciprocal condition number. Compared with the threshold

sqrt(n) * slamch('Epsilon') to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*A, where S scales each row by a power of the

radix so all absolute row sums of Z are approximately 1.

This subroutine is only responsible for setting the second field

above.

See Lapack Working Note 165 for further details and extra

cautions.

*ERR_BNDS_COMP*

ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

abs(XTRUE(j,i) - X(j,i))

max_j ----------------------

abs(X(j,i))

The array is indexed by the right-hand side i (on which the

componentwise relative error depends), and the type of error

information as described below. There currently are up to three

pieces of information returned for each right-hand side. If

componentwise accuracy is not requested (PARAMS(3) = 0.0), then

ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most

the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith

right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following

three fields:

err = 1 "Trust/don't trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * slamch('Epsilon').

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * slamch('Epsilon'). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated componentwise

reciprocal condition number. Compared with the threshold

sqrt(n) * slamch('Epsilon') to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*(A*diag(x)), where x is the solution for the

current right-hand side and S scales each row of

A*diag(x) by a power of the radix so all absolute row

sums of Z are approximately 1.

This subroutine is only responsible for setting the second field

above.

See Lapack Working Note 165 for further details and extra

cautions.

*RES*

RES is REAL array, dimension (N)

Workspace to hold the intermediate residual.

*AYB*

AYB is REAL array, dimension (N)

Workspace. This can be the same workspace passed for Y_TAIL.

*DY*

DY is REAL array, dimension (N)

Workspace to hold the intermediate solution.

*Y_TAIL*

Y_TAIL is REAL array, dimension (N)

Workspace to hold the trailing bits of the intermediate solution.

*RCOND*

RCOND is REAL

Reciprocal scaled condition number. This is an estimate of the

reciprocal Skeel condition number of the matrix A after

equilibration (if done). If this is less than the machine

precision (in particular, if it is zero), the matrix is singular

to working precision. Note that the error may still be small even

if this number is very small and the matrix appears ill-

conditioned.

*ITHRESH*

ITHRESH is INTEGER

The maximum number of residual computations allowed for

refinement. The default is 10. For 'aggressive' set to 100 to

permit convergence using approximate factorizations or

factorizations other than LU. If the factorization uses a

technique other than Gaussian elimination, the guarantees in

ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.

*RTHRESH*

RTHRESH is REAL

Determines when to stop refinement if the error estimate stops

decreasing. Refinement will stop when the next solution no longer

satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is

the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The

default value is 0.5. For 'aggressive' set to 0.9 to permit

convergence on extremely ill-conditioned matrices. See LAWN 165

for more details.

*DZ_UB*

DZ_UB is REAL

Determines when to start considering componentwise convergence.

Componentwise convergence is only considered after each component

of the solution Y is stable, which we define as the relative

change in each component being less than DZ_UB. The default value

is 0.25, requiring the first bit to be stable. See LAWN 165 for

more details.

*IGNORE_CWISE*

IGNORE_CWISE is LOGICAL

If .TRUE. then ignore componentwise convergence. Default value

is .FALSE..

*INFO*

INFO is INTEGER

= 0: Successful exit.

< 0: if INFO = -i, the ith argument to SLA_SYRFSX_EXTENDED had an illegal

value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## real function sla_syrpvgrw (character*1 UPLO, integer N, integer INFO, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) WORK)¶

**SLA_SYRPVGRW** computes the reciprocal pivot growth factor
norm(A)/norm(U) for a symmetric indefinite matrix.

**Purpose:**

SLA_SYRPVGRW computes the reciprocal pivot growth factor

norm(A)/norm(U). The "max absolute element" norm is used. If this is

much less than 1, the stability of the LU factorization of the

(equilibrated) matrix A could be poor. This also means that the

solution X, estimated condition numbers, and error bounds could be

unreliable.

**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 number of linear equations, i.e., the order of the

matrix A. N >= 0.

*INFO*

INFO is INTEGER

The value of INFO returned from SSYTRF, .i.e., the pivot in

column INFO is exactly 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the N-by-N matrix A.

*LDA*

LDA is INTEGER

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

*AF*

AF is REAL array, dimension (LDAF,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*WORK*

WORK is REAL array, dimension (2*N)

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine slasyf (character UPLO, integer N, integer NB, integer KB, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldw, * ) W, integer LDW, integer INFO)¶

**SLASYF** computes a partial factorization of a real symmetric
matrix using the Bunch-Kaufman diagonal pivoting method.

**Purpose:**

SLASYF computes a partial factorization of a real symmetric matrix A

using the Bunch-Kaufman diagonal pivoting method. The partial

factorization has the form:

A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:

( 0 U22 ) ( 0 D ) ( U12**T U22**T )

A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'

( L21 I ) ( 0 A22 ) ( 0 I )

where the order of D is at most NB. The actual order is returned in

the argument KB, and is either NB or NB-1, or N if N <= NB.

SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code

(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or

A22 (if UPLO = 'L').

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is stored:

= 'U': Upper triangular

= 'L': Lower triangular

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NB*

NB is INTEGER

The maximum number of columns of the matrix A that should be

factored. NB should be at least 2 to allow for 2-by-2 pivot

blocks.

*KB*

KB is INTEGER

The number of columns of A that were actually factored.

KB is either NB-1 or NB, or N if N <= NB.

*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, A contains details of the partial factorization.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If UPLO = 'U':

Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were

interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns

k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)

is a 2-by-2 diagonal block.

If UPLO = 'L':

Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were

interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns

k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)

is a 2-by-2 diagonal block.

*W*

W is REAL array, dimension (LDW,NB)

*LDW*

LDW is INTEGER

The leading dimension of the array W. LDW >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

> 0: if INFO = k, D(k,k) is exactly zero. The factorization

has been completed, but the block diagonal matrix D is

exactly singular.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2013, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

## subroutine slasyf_aa (character UPLO, integer J1, integer M, integer NB, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WORK)¶

**SLASYF_AA**

**Purpose:**

DLATRF_AA factorizes a panel of a real symmetric matrix A using

the Aasen's algorithm. The panel consists of a set of NB rows of A

when UPLO is U, or a set of NB columns when UPLO is L.

In order to factorize the panel, the Aasen's algorithm requires the

last row, or column, of the previous panel. The first row, or column,

of A is set to be the first row, or column, of an identity matrix,

which is used to factorize the first panel.

The resulting J-th row of U, or J-th column of L, is stored in the

(J-1)-th row, or column, of A (without the unit diagonals), while

the diagonal and subdiagonal of A are overwritten by those of T.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*J1*

J1 is INTEGER

The location of the first row, or column, of the panel

within the submatrix of A, passed to this routine, e.g.,

when called by SSYTRF_AA, for the first panel, J1 is 1,

while for the remaining panels, J1 is 2.

*M*

M is INTEGER

The dimension of the submatrix. M >= 0.

*NB*

NB is INTEGER

The dimension of the panel to be facotorized.

*A*

A is REAL array, dimension (LDA,M) for

the first panel, while dimension (LDA,M+1) for the

remaining panels.

On entry, A contains the last row, or column, of

the previous panel, and the trailing submatrix of A

to be factorized, except for the first panel, only

the panel is passed.

On exit, the leading panel is factorized.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*IPIV*

IPIV is INTEGER array, dimension (M)

Details of the row and column interchanges,

the row and column k were interchanged with the row and

column IPIV(k).

*H*

H is REAL workspace, dimension (LDH,NB).

*LDH*

LDH is INTEGER

The leading dimension of the workspace H. LDH >= max(1,M).

*WORK*

WORK is REAL workspace, dimension (M).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine slasyf_rook (character UPLO, integer N, integer NB, integer KB, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldw, * ) W, integer LDW, integer INFO)¶

**SLASYF_ROOK** computes a partial factorization of a real
symmetric matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting
method.

**Purpose:**

SLASYF_ROOK computes a partial factorization of a real symmetric

matrix A using the bounded Bunch-Kaufman ("rook") diagonal

pivoting method. The partial factorization has the form:

A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:

( 0 U22 ) ( 0 D ) ( U12**T U22**T )

A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'

( L21 I ) ( 0 A22 ) ( 0 I )

where the order of D is at most NB. The actual order is returned in

the argument KB, and is either NB or NB-1, or N if N <= NB.

SLASYF_ROOK is an auxiliary routine called by SSYTRF_ROOK. It uses

blocked code (calling Level 3 BLAS) to update the submatrix

A11 (if UPLO = 'U') or A22 (if UPLO = 'L').

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is stored:

= 'U': Upper triangular

= 'L': Lower triangular

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NB*

NB is INTEGER

The maximum number of columns of the matrix A that should be

factored. NB should be at least 2 to allow for 2-by-2 pivot

blocks.

*KB*

KB is INTEGER

The number of columns of A that were actually factored.

KB is either NB-1 or NB, or N if N <= NB.

*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, A contains details of the partial factorization.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If UPLO = 'U':

Only the last KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were

interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and

columns k and -IPIV(k) were interchanged and rows and

columns k-1 and -IPIV(k-1) were inerchaged,

D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = 'L':

Only the first KB elements of IPIV are set.

If IPIV(k) > 0, then rows and columns k and IPIV(k)

were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and

columns k and -IPIV(k) were interchanged and rows and

columns k+1 and -IPIV(k+1) were inerchaged,

D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

*W*

W is REAL array, dimension (LDW,NB)

*LDW*

LDW is INTEGER

The leading dimension of the array W. LDW >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

> 0: if INFO = k, D(k,k) is exactly zero. The factorization

has been completed, but the block diagonal matrix D is

exactly singular.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

November 2013, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,

School of Mathematics,

University of Manchester

## subroutine ssycon (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SSYCON**

**Purpose:**

SSYCON estimates the reciprocal of the condition number (in the

1-norm) of a real symmetric matrix A using the factorization

A = U*D*U**T or A = L*D*L**T computed by SSYTRF.

An estimate is obtained for norm(inv(A)), and the reciprocal of the

condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*ANORM*

ANORM is REAL

The 1-norm of the original matrix A.

*RCOND*

RCOND is REAL

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an

estimate of the 1-norm of inv(A) computed in this routine.

*WORK*

WORK is REAL array, dimension (2*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssycon_rook (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** SSYCON_ROOK **

**Purpose:**

SSYCON_ROOK estimates the reciprocal of the condition number (in the

1-norm) of a real symmetric matrix A using the factorization

A = U*D*U**T or A = L*D*L**T computed by SSYTRF_ROOK.

An estimate is obtained for norm(inv(A)), and the reciprocal of the

condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF_ROOK.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF_ROOK.

*ANORM*

ANORM is REAL

The 1-norm of the original matrix A.

*RCOND*

RCOND is REAL

The reciprocal of the condition number of the matrix A,

computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an

estimate of the 1-norm of inv(A) computed in this routine.

*WORK*

WORK is REAL array, dimension (2*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

December 2016, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,

School of Mathematics,

University of Manchester

## subroutine ssyconv (character UPLO, character WAY, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) E, integer INFO)¶

**SSYCONV**

**Purpose:**

SSYCONV convert A given by TRF into L and D and vice-versa.

Get Non-diag elements of D (returned in workspace) and

apply or reverse permutation done in TRF.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*WAY*

WAY is CHARACTER*1

= 'C': Convert

= 'R': Revert

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*E*

E is REAL array, dimension (N)

E stores the supdiagonal/subdiagonal of the symmetric 1-by-1

or 2-by-2 block diagonal matrix D in LDLT.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssyequb (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, real, dimension( * ) WORK, integer INFO)¶

**SSYEQUB**

**Purpose:**

SSYEQUB computes row and column scalings intended to equilibrate a

symmetric matrix A (with respect to the Euclidean norm) and reduce

its condition number. The scale factors S are computed by the BIN

algorithm (see references) so that the scaled matrix B with elements

B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of

the smallest possible condition number over all possible diagonal

scalings.

**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)

The N-by-N symmetric matrix whose scaling factors are to be

computed.

*LDA*

LDA is INTEGER

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

*S*

S is REAL array, dimension (N)

If INFO = 0, S contains the scale factors for A.

*SCOND*

SCOND is REAL

If INFO = 0, S contains the ratio of the smallest S(i) to

the largest S(i). If SCOND >= 0.1 and AMAX is neither too

large nor too small, it is not worth scaling by S.

*AMAX*

AMAX is REAL

Largest absolute value of any matrix element. If AMAX is

very close to overflow or very close to underflow, the

matrix should be scaled.

*WORK*

WORK is REAL array, dimension (2*N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, the i-th diagonal element is nonpositive.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**References:**

Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004.

DOI 10.1023/B:NUMA.0000016606.32820.69

Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679

## subroutine ssygs2 (integer ITYPE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**SSYGS2** reduces a symmetric definite generalized
eigenproblem to standard form, using the factorization results obtained from
spotrf (unblocked algorithm).

**Purpose:**

SSYGS2 reduces a real symmetric-definite generalized eigenproblem

to standard form.

If ITYPE = 1, the problem is A*x = lambda*B*x,

and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)

If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or

B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T *A*L.

B must have been previously factorized as U**T *U or L*L**T by SPOTRF.

**Parameters**

*ITYPE*

ITYPE is INTEGER

= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);

= 2 or 3: compute U*A*U**T or L**T *A*L.

*UPLO*

UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is stored, and how B has been factorized.

= 'U': Upper triangular

= 'L': Lower triangular

*N*

N is INTEGER

The order of the matrices A and B. 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 INFO = 0, the transformed matrix, stored in the

same format as A.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB,N)

The triangular factor from the Cholesky factorization of B,

as returned by SPOTRF.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssygst (integer ITYPE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**SSYGST**

**Purpose:**

SSYGST reduces a real symmetric-definite generalized eigenproblem

to standard form.

If ITYPE = 1, the problem is A*x = lambda*B*x,

and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)

If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or

B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.

B must have been previously factorized as U**T*U or L*L**T by SPOTRF.

**Parameters**

*ITYPE*

ITYPE is INTEGER

= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);

= 2 or 3: compute U*A*U**T or L**T*A*L.

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored and B is factored as

U**T*U;

= 'L': Lower triangle of A is stored and B is factored as

L*L**T.

*N*

N is INTEGER

The order of the matrices A and B. 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 INFO = 0, the transformed matrix, stored in the

same format as A.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB,N)

The triangular factor from the Cholesky factorization of B,

as returned by SPOTRF.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssyrfs (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SSYRFS**

**Purpose:**

SSYRFS improves the computed solution to a system of linear

equations when the coefficient matrix is symmetric indefinite, and

provides error bounds and backward error estimates for the solution.

**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.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrices B and X. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

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.

*LDA*

LDA is INTEGER

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

*AF*

AF is REAL array, dimension (LDAF,N)

The factored form of the matrix A. AF contains the block

diagonal matrix D and the multipliers used to obtain the

factor U or L from the factorization A = U*D*U**T or

A = L*D*L**T as computed by SSYTRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*B*

B is REAL array, dimension (LDB,NRHS)

The right hand side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is REAL array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by SSYTRS.

On exit, the improved solution matrix X.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*FERR*

FERR is REAL array, dimension (NRHS)

The estimated forward error bound for each solution vector

X(j) (the j-th column of the solution matrix X).

If XTRUE is the true solution corresponding to X(j), FERR(j)

is an estimated upper bound for the magnitude of the largest

element in (X(j) - XTRUE) divided by the magnitude of the

largest element in X(j). The estimate is as reliable as

the estimate for RCOND, and is almost always a slight

overestimate of the true error.

*BERR*

BERR is REAL array, dimension (NRHS)

The componentwise relative backward error of each solution

vector X(j) (i.e., the smallest relative change in

any element of A or B that makes X(j) an exact solution).

*WORK*

WORK is REAL array, dimension (3*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Internal Parameters:**

ITMAX is the maximum number of steps of iterative refinement.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssyrfsx (character UPLO, character EQUED, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

**SSYRFSX**

**Purpose:**

SSYRFSX improves the computed solution to a system of linear

equations when the coefficient matrix is symmetric indefinite, and

provides error bounds and backward error estimates for the

solution. In addition to normwise error bound, the code provides

maximum componentwise error bound if possible. See comments for

ERR_BNDS_NORM and ERR_BNDS_COMP for details of the error bounds.

The original system of linear equations may have been equilibrated

before calling this routine, as described by arguments EQUED and S

below. In this case, the solution and error bounds returned are

for the original unequilibrated system.

Some optional parameters are bundled in the PARAMS array. These

settings determine how refinement is performed, but often the

defaults are acceptable. If the defaults are acceptable, users

can pass NPARAMS = 0 which prevents the source code from accessing

the PARAMS argument.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;

= 'L': Lower triangle of A is stored.

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done to A

before calling this routine. This is needed to compute

the solution and error bounds correctly.

= 'N': No equilibration

= 'Y': Both row and column equilibration, i.e., A has been

replaced by diag(S) * A * diag(S).

The right hand side B has been changed accordingly.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrices B and X. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

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.

*LDA*

LDA is INTEGER

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

*AF*

AF is REAL array, dimension (LDAF,N)

The factored form of the matrix A. AF contains the block

diagonal matrix D and the multipliers used to obtain the

factor U or L from the factorization A = U*D*U**T or A =

L*D*L**T as computed by SSYTRF.

*LDAF*

LDAF is INTEGER

The leading dimension of the array AF. LDAF >= max(1,N).

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*S*

S is REAL array, dimension (N)

The scale factors for A. If EQUED = 'Y', A is multiplied on

the left and right by diag(S). S is an input argument if FACT =

'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED

= 'Y', each element of S must be positive. If S is output, each

element of S is a power of the radix. If S is input, each element

of S should be a power of the radix to ensure a reliable solution

and error estimates. Scaling by powers of the radix does not cause

rounding errors unless the result underflows or overflows.

Rounding errors during scaling lead to refining with a matrix that

is not equivalent to the input matrix, producing error estimates

that may not be reliable.

*B*

B is REAL array, dimension (LDB,NRHS)

The right hand side matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*X*

X is REAL array, dimension (LDX,NRHS)

On entry, the solution matrix X, as computed by SGETRS.

On exit, the improved solution matrix X.

*LDX*

LDX is INTEGER

The leading dimension of the array X. LDX >= max(1,N).

*RCOND*

RCOND is REAL

Reciprocal scaled condition number. This is an estimate of the

reciprocal Skeel condition number of the matrix A after

equilibration (if done). If this is less than the machine

precision (in particular, if it is zero), the matrix is singular

to working precision. Note that the error may still be small even

if this number is very small and the matrix appears ill-

conditioned.

*BERR*

BERR is REAL array, dimension (NRHS)

Componentwise relative backward error. This is the

componentwise relative backward error of each solution vector X(j)

(i.e., the smallest relative change in any element of A or B that

makes X(j) an exact solution).

*N_ERR_BNDS*

N_ERR_BNDS is INTEGER

Number of error bounds to return for each right hand side

and each type (normwise or componentwise). See ERR_BNDS_NORM and

ERR_BNDS_COMP below.

*ERR_BNDS_NORM*

ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

normwise relative error, which is defined as follows:

Normwise relative error in the ith solution vector:

max_j (abs(XTRUE(j,i) - X(j,i)))

------------------------------

max_j abs(X(j,i))

The array is indexed by the type of error information as described

below. There currently are up to three pieces of information

returned.

The first index in ERR_BNDS_NORM(i,:) corresponds to the ith

right-hand side.

The second index in ERR_BNDS_NORM(:,err) contains the following

three fields:

err = 1 "Trust/don't trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * slamch('Epsilon').

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * slamch('Epsilon'). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated normwise

reciprocal condition number. Compared with the threshold

sqrt(n) * slamch('Epsilon') to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*A, where S scales each row by a power of the

radix so all absolute row sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra

cautions.

*ERR_BNDS_COMP*

ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)

For each right-hand side, this array contains information about

various error bounds and condition numbers corresponding to the

componentwise relative error, which is defined as follows:

Componentwise relative error in the ith solution vector:

abs(XTRUE(j,i) - X(j,i))

max_j ----------------------

abs(X(j,i))

The array is indexed by the right-hand side i (on which the

componentwise relative error depends), and the type of error

information as described below. There currently are up to three

pieces of information returned for each right-hand side. If

componentwise accuracy is not requested (PARAMS(3) = 0.0), then

ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most

the first (:,N_ERR_BNDS) entries are returned.

The first index in ERR_BNDS_COMP(i,:) corresponds to the ith

right-hand side.

The second index in ERR_BNDS_COMP(:,err) contains the following

three fields:

err = 1 "Trust/don't trust" boolean. Trust the answer if the

reciprocal condition number is less than the threshold

sqrt(n) * slamch('Epsilon').

err = 2 "Guaranteed" error bound: The estimated forward error,

almost certainly within a factor of 10 of the true error

so long as the next entry is greater than the threshold

sqrt(n) * slamch('Epsilon'). This error bound should only

be trusted if the previous boolean is true.

err = 3 Reciprocal condition number: Estimated componentwise

reciprocal condition number. Compared with the threshold

sqrt(n) * slamch('Epsilon') to determine if the error

estimate is "guaranteed". These reciprocal condition

numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some

appropriately scaled matrix Z.

Let Z = S*(A*diag(x)), where x is the solution for the

current right-hand side and S scales each row of

A*diag(x) by a power of the radix so all absolute row

sums of Z are approximately 1.

See Lapack Working Note 165 for further details and extra

cautions.

*NPARAMS*

NPARAMS is INTEGER

Specifies the number of parameters set in PARAMS. If <= 0, the

PARAMS array is never referenced and default values are used.

*PARAMS*

PARAMS is REAL array, dimension NPARAMS

Specifies algorithm parameters. If an entry is < 0.0, then

that entry will be filled with default value used for that

parameter. Only positions up to NPARAMS are accessed; defaults

are used for higher-numbered parameters.

PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative

refinement or not.

Default: 1.0

= 0.0: No refinement is performed, and no error bounds are

computed.

= 1.0: Use the double-precision refinement algorithm,

possibly with doubled-single computations if the

compilation environment does not support DOUBLE

PRECISION.

(other values are reserved for future use)

PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual

computations allowed for refinement.

Default: 10

Aggressive: Set to 100 to permit convergence using approximate

factorizations or factorizations other than LU. If

the factorization uses a technique other than

Gaussian elimination, the guarantees in

err_bnds_norm and err_bnds_comp may no longer be

trustworthy.

PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code

will attempt to find a solution with small componentwise

relative error in the double-precision algorithm. Positive

is true, 0.0 is false.

Default: 1.0 (attempt componentwise convergence)

*WORK*

WORK is REAL array, dimension (4*N)

*IWORK*

IWORK is INTEGER array, dimension (N)

*INFO*

INFO is INTEGER

= 0: Successful exit. The solution to every right-hand side is

guaranteed.

< 0: If INFO = -i, the i-th argument had an illegal value

> 0 and <= N: U(INFO,INFO) is exactly zero. The factorization

has been completed, but the factor U is exactly singular, so

the solution and error bounds could not be computed. RCOND = 0

is returned.

= N+J: The solution corresponding to the Jth right-hand side is

not guaranteed. The solutions corresponding to other right-

hand sides K with K > J may not be guaranteed as well, but

only the first such right-hand side is reported. If a small

componentwise error is not requested (PARAMS(3) = 0.0) then

the Jth right-hand side is the first with a normwise error

bound that is not guaranteed (the smallest J such

that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)

the Jth right-hand side is the first with either a normwise or

componentwise error bound that is not guaranteed (the smallest

J such that either ERR_BNDS_NORM(J,1) = 0.0 or

ERR_BNDS_COMP(J,1) = 0.0). See the definition of

ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information

about all of the right-hand sides check ERR_BNDS_NORM or

ERR_BNDS_COMP.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytd2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAU, integer INFO)¶

**SSYTD2** reduces a symmetric matrix to real symmetric
tridiagonal form by an orthogonal similarity transformation (unblocked
algorithm).

**Purpose:**

SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal

form T by an orthogonal similarity transformation: Q**T * A * Q = T.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is stored:

= 'U': Upper triangular

= 'L': Lower triangular

*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).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

**Author**

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).

## subroutine ssytf2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶

**SSYTF2** computes the factorization of a real symmetric
indefinite matrix, using the diagonal pivoting method (unblocked
algorithm).

**Purpose:**

SSYTF2 computes the factorization of a real symmetric matrix A using

the Bunch-Kaufman diagonal pivoting method:

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)

triangular matrices, U**T is the transpose of U, and D is symmetric and

block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is stored:

= 'U': Upper triangular

= 'L': Lower triangular

*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, the block diagonal matrix D and the multipliers used

to obtain the factor U or L (see below for further details).

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If UPLO = 'U':

If IPIV(k) > 0, then rows and columns k and IPIV(k) were

interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k-1) < 0, then rows and columns

k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)

is a 2-by-2 diagonal block.

If UPLO = 'L':

If IPIV(k) > 0, then rows and columns k and IPIV(k) were

interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) = IPIV(k+1) < 0, then rows and columns

k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)

is a 2-by-2 diagonal block.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -k, the k-th argument had an illegal value

> 0: if INFO = k, D(k,k) is exactly zero. The factorization

has been completed, but the block diagonal matrix D is

exactly singular, and division by zero will occur if it

is used to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

If UPLO = 'U', then A = U*D*U**T, where

U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to

1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I v 0 ) k-s

U(k) = ( 0 I 0 ) s

( 0 0 I ) n-k

k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),

and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I 0 0 ) k-1

L(k) = ( 0 I 0 ) s

( 0 v I ) n-k-s+1

k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).

If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),

and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

**Contributors:**

09-29-06 - patch from

Bobby Cheng, MathWorks

Replace l.204 and l.372

IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN

by

IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. SISNAN(ABSAKK) ) THEN

01-01-96 - Based on modifications by

J. Lewis, Boeing Computer Services Company

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

1-96 - Based on modifications by J. Lewis, Boeing Computer Services

Company

## subroutine ssytf2_rook (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, integer INFO)¶

**SSYTF2_ROOK** computes the factorization of a real symmetric
indefinite matrix using the bounded Bunch-Kaufman ('rook') diagonal pivoting
method (unblocked algorithm).

**Purpose:**

SSYTF2_ROOK computes the factorization of a real symmetric matrix A

using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)

triangular matrices, U**T is the transpose of U, and D is symmetric and

block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is stored:

= 'U': Upper triangular

= 'L': Lower triangular

*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, the block diagonal matrix D and the multipliers used

to obtain the factor U or L (see below for further details).

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If UPLO = 'U':

If IPIV(k) > 0, then rows and columns k and IPIV(k)

were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and

columns k and -IPIV(k) were interchanged and rows and

columns k-1 and -IPIV(k-1) were inerchaged,

D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = 'L':

If IPIV(k) > 0, then rows and columns k and IPIV(k)

were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and

columns k and -IPIV(k) were interchanged and rows and

columns k+1 and -IPIV(k+1) were inerchaged,

D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -k, the k-th argument had an illegal value

> 0: if INFO = k, D(k,k) is exactly zero. The factorization

has been completed, but the block diagonal matrix D is

exactly singular, and division by zero will occur if it

is used to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

If UPLO = 'U', then A = U*D*U**T, where

U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to

1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I v 0 ) k-s

U(k) = ( 0 I 0 ) s

( 0 0 I ) n-k

k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),

and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I 0 0 ) k-1

L(k) = ( 0 I 0 ) s

( 0 v I ) n-k-s+1

k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).

If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),

and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

**Contributors:**

November 2013, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,

School of Mathematics,

University of Manchester

01-01-96 - Based on modifications by

J. Lewis, Boeing Computer Services Company

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

## 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 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).

## subroutine ssytrd_2stage (character VECT, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAU, real, dimension( * ) HOUS2, integer LHOUS2, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYTRD_2STAGE**

**Purpose:**

SSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric

tridiagonal form T by a orthogonal similarity transformation:

Q1**T Q2**T* A * Q2 * Q1 = T.

**Parameters**

*VECT*

VECT is CHARACTER*1

= 'N': No need for the Housholder representation,

in particular for the second stage (Band to

tridiagonal) and thus LHOUS2 is of size max(1, 4*N);

= 'V': the Householder representation is needed to

either generate Q1 Q2 or to apply Q1 Q2,

then LHOUS2 is to be queried and computed.

(NOT AVAILABLE IN THIS RELEASE).

*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 band superdiagonal

of A are overwritten by the corresponding elements of the

internal band-diagonal matrix AB, and the elements above

the KD superdiagonal, with the array TAU, represent the orthogonal

matrix Q1 as a product of elementary reflectors; if UPLO

= 'L', the diagonal and band subdiagonal of A are over-

written by the corresponding elements of the internal band-diagonal

matrix AB, and the elements below the KD subdiagonal, with

the array TAU, represent the orthogonal matrix Q1 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.

*E*

E is REAL array, dimension (N-1)

The off-diagonal elements of the tridiagonal matrix T.

*TAU*

TAU is REAL array, dimension (N-KD)

The scalar factors of the elementary reflectors of

the first stage (see Further Details).

*HOUS2*

HOUS2 is REAL array, dimension (LHOUS2)

Stores the Householder representation of the stage2

band to tridiagonal.

*LHOUS2*

LHOUS2 is INTEGER

The dimension of the array HOUS2.

If LWORK = -1, or LHOUS2 = -1,

then a query is assumed; the routine

only calculates the optimal size of the HOUS2 array, returns

this value as the first entry of the HOUS2 array, and no error

message related to LHOUS2 is issued by XERBLA.

If VECT='N', LHOUS2 = max(1, 4*n);

if VECT='V', option not yet available.

*WORK*

WORK is REAL array, dimension (LWORK)

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK = MAX(1, dimension)

If LWORK = -1, or LHOUS2=-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.

LWORK = MAX(1, dimension) where

dimension = max(stage1,stage2) + (KD+1)*N

= N*KD + N*max(KD+1,FACTOPTNB)

+ max(2*KD*KD, KD*NTHREADS)

+ (KD+1)*N

where KD is the blocking size of the reduction,

FACTOPTNB is the blocking used by the QR or LQ

algorithm, usually FACTOPTNB=128 is a good choice

NTHREADS is the number of threads used when

openMP compilation is enabled, otherwise =1.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.

Parallel reduction to condensed forms for symmetric eigenvalue problems

using aggregated fine-grained and memory-aware kernels. In Proceedings

of 2011 International Conference for High Performance Computing,

Networking, Storage and Analysis (SC '11), New York, NY, USA,

Article 8 , 11 pages.

http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.

An improved parallel singular value algorithm and its implementation

for multicore hardware, In Proceedings of 2013 International Conference

for High Performance Computing, Networking, Storage and Analysis (SC '13).

Denver, Colorado, USA, 2013.

Article 90, 12 pages.

http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.

A novel hybrid CPU-GPU generalized eigensolver for electronic structure

calculations based on fine-grained memory aware tasks.

International Journal of High Performance Computing Applications.

Volume 28 Issue 2, Pages 196-209, May 2014.

http://hpc.sagepub.com/content/28/2/196

## subroutine ssytrd_sy2sb (character UPLO, integer N, integer KD, real, dimension( lda, * ) A, integer LDA, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) TAU, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYTRD_SY2SB**

**Purpose:**

SSYTRD_SY2SB reduces a real symmetric matrix A to real symmetric

band-diagonal form AB by a orthogonal similarity transformation:

Q**T * A * Q = AB.

**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.

*KD*

KD is INTEGER

The number of superdiagonals of the reduced matrix if UPLO = 'U',

or the number of subdiagonals if UPLO = 'L'. KD >= 0.

The reduced matrix is stored in the array AB.

*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).

*AB*

AB is REAL array, dimension (LDAB,N)

On exit, the upper or lower triangle of the symmetric band

matrix A, stored in the first KD+1 rows of the array. The

j-th column of A is stored in the j-th column of the array AB

as follows:

if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;

if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).

*LDAB*

LDAB is INTEGER

The leading dimension of the array AB. LDAB >= KD+1.

*TAU*

TAU is REAL array, dimension (N-KD)

The scalar factors of the elementary reflectors (see Further

Details).

*WORK*

WORK is REAL array, dimension (LWORK)

On exit, if INFO = 0, or if LWORK=-1,

WORK(1) returns the size of LWORK.

*LWORK*

LWORK is INTEGER

The dimension of the array WORK which should be calculated

by a workspace query. LWORK = MAX(1, LWORK_QUERY)

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.

LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD

where FACTOPTNB is the blocking used by the QR or LQ

algorithm, usually FACTOPTNB=128 is a good choice otherwise

putting LWORK=-1 will provide the size of WORK.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Implemented by Azzam Haidar.

All details are available on technical report, SC11, SC13 papers.

Azzam Haidar, Hatem Ltaief, and Jack Dongarra.

Parallel reduction to condensed forms for symmetric eigenvalue problems

using aggregated fine-grained and memory-aware kernels. In Proceedings

of 2011 International Conference for High Performance Computing,

Networking, Storage and Analysis (SC '11), New York, NY, USA,

Article 8 , 11 pages.

http://doi.acm.org/10.1145/2063384.2063394

A. Haidar, J. Kurzak, P. Luszczek, 2013.

An improved parallel singular value algorithm and its implementation

for multicore hardware, In Proceedings of 2013 International Conference

for High Performance Computing, Networking, Storage and Analysis (SC '13).

Denver, Colorado, USA, 2013.

Article 90, 12 pages.

http://doi.acm.org/10.1145/2503210.2503292

A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.

A novel hybrid CPU-GPU generalized eigensolver for electronic structure

calculations based on fine-grained memory aware tasks.

International Journal of High Performance Computing Applications.

Volume 28 Issue 2, Pages 196-209, May 2014.

http://hpc.sagepub.com/content/28/2/196

If UPLO = 'U', the matrix Q is represented as a product of elementary

reflectors

Q = H(k)**T . . . H(2)**T H(1)**T, where k = n-kd.

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+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in

A(i,i+kd+1:n), 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(k), where k = n-kd.

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(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in

A(i+kd+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':

( ab ab/v1 v1 v1 v1 ) ( ab )

( ab ab/v2 v2 v2 ) ( ab/v1 ab )

( ab ab/v3 v3 ) ( v1 ab/v2 ab )

( ab ab/v4 ) ( v1 v2 ab/v3 ab )

( ab ) ( v1 v2 v3 ab/v4 ab )

where d and e denote diagonal and off-diagonal elements of T, and vi

denotes an element of the vector defining H(i)..fi

## subroutine ssytrf (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYTRF**

**Purpose:**

SSYTRF computes the factorization of a real symmetric matrix A using

the Bunch-Kaufman diagonal pivoting method. The form of the

factorization is

A = U**T*D*U or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)

triangular matrices, and D is symmetric and block diagonal with

1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.

**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, the block diagonal matrix D and the multipliers used

to obtain the factor U or L (see below for further details).

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If IPIV(k) > 0, then rows and columns k and IPIV(k) were

interchanged and D(k,k) is a 1-by-1 diagonal block.

If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and

columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)

is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =

IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were

interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

*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 length of WORK. LWORK >=1. For best performance

LWORK >= N*NB, where NB is the block size returned by ILAENV.

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

> 0: if INFO = i, D(i,i) is exactly zero. The factorization

has been completed, but the block diagonal matrix D is

exactly singular, and division by zero will occur if it

is used to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

If UPLO = 'U', then A = U**T*D*U, where

U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to

1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I v 0 ) k-s

U(k) = ( 0 I 0 ) s

( 0 0 I ) n-k

k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),

and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I 0 0 ) k-1

L(k) = ( 0 I 0 ) s

( 0 v I ) n-k-s+1

k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).

If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),

and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

## subroutine ssytrf_aa (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYTRF_AA**

**Purpose:**

SSYTRF_AA computes the factorization of a real symmetric matrix A

using the Aasen's algorithm. The form of the factorization is

A = U**T*T*U or A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower)

triangular matrices, and T is a symmetric tridiagonal matrix.

This is the blocked version of the algorithm, calling Level 3 BLAS.

**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, the tridiagonal matrix is stored in the diagonals

and the subdiagonals of A just below (or above) the diagonals,

and L is stored below (or above) the subdiaonals, when UPLO

is 'L' (or 'U').

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

On exit, it contains the details of the interchanges, i.e.,

the row and column k of A were interchanged with the

row and column IPIV(k).

*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 length of WORK. LWORK >= MAX(1,2*N). For optimum performance

LWORK >= N*(1+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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytrf_aa_2stage (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYTRF_AA_2STAGE**

**Purpose:**

SSYTRF_AA_2STAGE computes the factorization of a real symmetric matrix A

using the Aasen's algorithm. The form of the factorization is

A = U**T*T*U or A = L*T*L**T

where U (or L) is a product of permutation and unit upper (lower)

triangular matrices, and T is a symmetric band matrix with the

bandwidth of NB (NB is internally selected and stored in TB( 1 ), and T is

LU factorized with partial pivoting).

This is the blocked version of the algorithm, calling Level 3 BLAS.

**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, L is stored below (or above) the subdiaonal blocks,

when UPLO is 'L' (or 'U').

*LDA*

LDA is INTEGER

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

*TB*

TB is REAL array, dimension (LTB)

On exit, details of the LU factorization of the band matrix.

*LTB*

LTB is INTEGER

The size of the array TB. LTB >= 4*N, internally

used to select NB such that LTB >= (3*NB+1)*N.

If LTB = -1, then a workspace query is assumed; the

routine only calculates the optimal size of LTB,

returns this value as the first entry of TB, and

no error message related to LTB is issued by XERBLA.

*IPIV*

IPIV is INTEGER array, dimension (N)

On exit, it contains the details of the interchanges, i.e.,

the row and column k of A were interchanged with the

row and column IPIV(k).

*IPIV2*

IPIV2 is INTEGER array, dimension (N)

On exit, it contains the details of the interchanges, i.e.,

the row and column k of T were interchanged with the

row and column IPIV(k).

*WORK*

WORK is REAL workspace of size LWORK

*LWORK*

LWORK is INTEGER

The size of WORK. LWORK >= N, internally used to select NB

such that LWORK >= N*NB.

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.

> 0: if INFO = i, band LU factorization failed on i-th column

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytrf_rook (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYTRF_ROOK**

**Purpose:**

SSYTRF_ROOK computes the factorization of a real symmetric matrix A

using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.

The form of the factorization is

A = U*D*U**T or A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower)

triangular matrices, and D is symmetric and block diagonal with

1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.

**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, the block diagonal matrix D and the multipliers used

to obtain the factor U or L (see below for further details).

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D.

If UPLO = 'U':

If IPIV(k) > 0, then rows and columns k and IPIV(k)

were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and

columns k and -IPIV(k) were interchanged and rows and

columns k-1 and -IPIV(k-1) were inerchaged,

D(k-1:k,k-1:k) is a 2-by-2 diagonal block.

If UPLO = 'L':

If IPIV(k) > 0, then rows and columns k and IPIV(k)

were interchanged and D(k,k) is a 1-by-1 diagonal block.

If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and

columns k and -IPIV(k) were interchanged and rows and

columns k+1 and -IPIV(k+1) were inerchaged,

D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

*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 length of WORK. LWORK >=1. For best performance

LWORK >= N*NB, where NB is the block size returned by ILAENV.

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

> 0: if INFO = i, D(i,i) is exactly zero. The factorization

has been completed, but the block diagonal matrix D is

exactly singular, and division by zero will occur if it

is used to solve a system of equations.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

If UPLO = 'U', then A = U*D*U**T, where

U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to

1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and U(k) is a unit upper triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I v 0 ) k-s

U(k) = ( 0 I 0 ) s

( 0 0 I ) n-k

k-s s n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).

If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),

and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to

n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1

and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as

defined by IPIV(k), and L(k) is a unit lower triangular matrix, such

that if the diagonal block D(k) is of order s (s = 1 or 2), then

( I 0 0 ) k-1

L(k) = ( 0 I 0 ) s

( 0 v I ) n-k-s+1

k-1 s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).

If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),

and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

**Contributors:**

June 2016, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,

School of Mathematics,

University of Manchester

## subroutine ssytri (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer INFO)¶

**SSYTRI**

**Purpose:**

SSYTRI computes the inverse of a real symmetric indefinite matrix

A using the factorization A = U*D*U**T or A = L*D*L**T computed by

SSYTRF.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the block diagonal matrix D and the multipliers

used to obtain the factor U or L as computed by SSYTRF.

On exit, if INFO = 0, the (symmetric) inverse of the original

matrix. If UPLO = 'U', the upper triangular part of the

inverse is formed and the part of A below the diagonal is not

referenced; if UPLO = 'L' the lower triangular part of the

inverse is formed and the part of A above the diagonal is

not referenced.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*WORK*

WORK is REAL array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

inverse could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytri2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYTRI2**

**Purpose:**

SSYTRI2 computes the inverse of a REAL symmetric indefinite matrix

A using the factorization A = U*D*U**T or A = L*D*L**T computed by

SSYTRF. SSYTRI2 sets the LEADING DIMENSION of the workspace

before calling SSYTRI2X that actually computes the inverse.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the block diagonal matrix D and the multipliers

used to obtain the factor U or L as computed by SSYTRF.

On exit, if INFO = 0, the (symmetric) inverse of the original

matrix. If UPLO = 'U', the upper triangular part of the

inverse is formed and the part of A below the diagonal is not

referenced; if UPLO = 'L' the lower triangular part of the

inverse is formed and the part of A above the diagonal is

not referenced.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*WORK*

WORK is REAL array, dimension (N+NB+1)*(NB+3)

*LWORK*

LWORK is INTEGER

The dimension of the array WORK.

WORK is size >= (N+NB+1)*(NB+3)

If LWORK = -1, then a workspace query is assumed; the routine

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

> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

inverse could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytri2x (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( n+nb+1,* ) WORK, integer NB, integer INFO)¶

**SSYTRI2X**

**Purpose:**

SSYTRI2X computes the inverse of a real symmetric indefinite matrix

A using the factorization A = U*D*U**T or A = L*D*L**T computed by

SSYTRF.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the NNB diagonal matrix D and the multipliers

used to obtain the factor U or L as computed by SSYTRF.

On exit, if INFO = 0, the (symmetric) inverse of the original

matrix. If UPLO = 'U', the upper triangular part of the

inverse is formed and the part of A below the diagonal is not

referenced; if UPLO = 'L' the lower triangular part of the

inverse is formed and the part of A above the diagonal is

not referenced.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the NNB structure of D

as determined by SSYTRF.

*WORK*

WORK is REAL array, dimension (N+NB+1,NB+3)

*NB*

NB is INTEGER

Block size

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

inverse could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytri_rook (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( * ) WORK, integer INFO)¶

**SSYTRI_ROOK**

**Purpose:**

SSYTRI_ROOK computes the inverse of a real symmetric

matrix A using the factorization A = U*D*U**T or A = L*D*L**T

computed by SSYTRF_ROOK.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*A*

A is REAL array, dimension (LDA,N)

On entry, the block diagonal matrix D and the multipliers

used to obtain the factor U or L as computed by SSYTRF_ROOK.

On exit, if INFO = 0, the (symmetric) inverse of the original

matrix. If UPLO = 'U', the upper triangular part of the

inverse is formed and the part of A below the diagonal is not

referenced; if UPLO = 'L' the lower triangular part of the

inverse is formed and the part of A above the diagonal is

not referenced.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF_ROOK.

*WORK*

WORK is REAL array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its

inverse could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

April 2012, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,

School of Mathematics,

University of Manchester

## subroutine ssytrs (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**SSYTRS**

**Purpose:**

SSYTRS solves a system of linear equations A*X = B with a real

symmetric matrix A using the factorization A = U*D*U**T or

A = L*D*L**T computed by SSYTRF.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the right hand side matrix B.

On exit, the solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytrs2 (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) WORK, integer INFO)¶

**SSYTRS2**

**Purpose:**

SSYTRS2 solves a system of linear equations A*X = B with a real

symmetric matrix A using the factorization A = U*D*U**T or

A = L*D*L**T computed by SSYTRF and converted by SSYCONV.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF.

Note that A is input / output. This might be counter-intuitive,

and one may think that A is input only. A is input / output. This

is because, at the start of the subroutine, we permute A in a

"better" form and then we permute A back to its original form at

the end.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the right hand side matrix B.

On exit, the solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*WORK*

WORK is REAL array, dimension (N)

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytrs_aa (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

**SSYTRS_AA**

**Purpose:**

SSYTRS_AA solves a system of linear equations A*X = B with a real

symmetric matrix A using the factorization A = U**T*T*U or

A = L*T*L**T computed by SSYTRF_AA.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U**T*T*U;

= 'L': Lower triangular, form is A = L*T*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

Details of factors computed by SSYTRF_AA.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges as computed by SSYTRF_AA.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the right hand side matrix B.

On exit, the solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*WORK*

WORK is REAL array, dimension (MAX(1,LWORK))

*LWORK*

LWORK is INTEGER

The dimension of the array WORK. LWORK >= max(1,3*N-2).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytrs_aa_2stage (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**SSYTRS_AA_2STAGE**

**Purpose:**

SSYTRS_AA_2STAGE solves a system of linear equations A*X = B with a real

symmetric matrix A using the factorization A = U**T*T*U or

A = L*T*L**T computed by SSYTRF_AA_2STAGE.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U**T*T*U;

= 'L': Lower triangular, form is A = L*T*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

Details of factors computed by SSYTRF_AA_2STAGE.

*LDA*

LDA is INTEGER

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

*TB*

TB is REAL array, dimension (LTB)

Details of factors computed by SSYTRF_AA_2STAGE.

*LTB*

LTB is INTEGER

The size of the array TB. LTB >= 4*N.

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges as computed by

SSYTRF_AA_2STAGE.

*IPIV2*

IPIV2 is INTEGER array, dimension (N)

Details of the interchanges as computed by

SSYTRF_AA_2STAGE.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the right hand side matrix B.

On exit, the solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## subroutine ssytrs_rook (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

**SSYTRS_ROOK**

**Purpose:**

SSYTRS_ROOK solves a system of linear equations A*X = B with

a real symmetric matrix A using the factorization A = U*D*U**T or

A = L*D*L**T computed by SSYTRF_ROOK.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the details of the factorization are stored

as an upper or lower triangular matrix.

= 'U': Upper triangular, form is A = U*D*U**T;

= 'L': Lower triangular, form is A = L*D*L**T.

*N*

N is INTEGER

The order of the matrix A. N >= 0.

*NRHS*

NRHS is INTEGER

The number of right hand sides, i.e., the number of columns

of the matrix B. NRHS >= 0.

*A*

A is REAL array, dimension (LDA,N)

The block diagonal matrix D and the multipliers used to

obtain the factor U or L as computed by SSYTRF_ROOK.

*LDA*

LDA is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D

as determined by SSYTRF_ROOK.

*B*

B is REAL array, dimension (LDB,NRHS)

On entry, the right hand side matrix B.

On exit, the solution matrix X.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

April 2012, Igor Kozachenko,

Computer Science Division,

University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,

School of Mathematics,

University of Manchester

## subroutine stgsyl (character TRANS, integer IJOB, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldc, * ) C, integer LDC, real, dimension( ldd, * ) D, integer LDD, real, dimension( lde, * ) E, integer LDE, real, dimension( ldf, * ) F, integer LDF, real SCALE, real DIF, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

**STGSYL**

**Purpose:**

STGSYL solves the generalized Sylvester equation:

A * R - L * B = scale * C (1)

D * R - L * E = scale * F

where R and L are unknown m-by-n matrices, (A, D), (B, E) and

(C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,

respectively, with real entries. (A, D) and (B, E) must be in

generalized (real) Schur canonical form, i.e. A, B are upper quasi

triangular and D, E are upper triangular.

The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output

scaling factor chosen to avoid overflow.

In matrix notation (1) is equivalent to solve Zx = scale b, where

Z is defined as

Z = [ kron(In, A) -kron(B**T, Im) ] (2)

[ kron(In, D) -kron(E**T, Im) ].

Here Ik is the identity matrix of size k and X**T is the transpose of

X. kron(X, Y) is the Kronecker product between the matrices X and Y.

If TRANS = 'T', STGSYL solves the transposed system Z**T*y = scale*b,

which is equivalent to solve for R and L in

A**T * R + D**T * L = scale * C (3)

R * B**T + L * E**T = scale * -F

This case (TRANS = 'T') is used to compute an one-norm-based estimate

of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)

and (B,E), using SLACON.

If IJOB >= 1, STGSYL computes a Frobenius norm-based estimate

of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the

reciprocal of the smallest singular value of Z. See [1-2] for more

information.

This is a level 3 BLAS algorithm.

**Parameters**

*TRANS*

TRANS is CHARACTER*1

= 'N': solve the generalized Sylvester equation (1).

= 'T': solve the 'transposed' system (3).

*IJOB*

IJOB is INTEGER

Specifies what kind of functionality to be performed.

= 0: solve (1) only.

= 1: The functionality of 0 and 3.

= 2: The functionality of 0 and 4.

= 3: Only an estimate of Dif[(A,D), (B,E)] is computed.

(look ahead strategy IJOB = 1 is used).

= 4: Only an estimate of Dif[(A,D), (B,E)] is computed.

( SGECON on sub-systems is used ).

Not referenced if TRANS = 'T'.

*M*

M is INTEGER

The order of the matrices A and D, and the row dimension of

the matrices C, F, R and L.

*N*

N is INTEGER

The order of the matrices B and E, and the column dimension

of the matrices C, F, R and L.

*A*

A is REAL array, dimension (LDA, M)

The upper quasi triangular matrix A.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1, M).

*B*

B is REAL array, dimension (LDB, N)

The upper quasi triangular matrix B.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1, N).

*C*

C is REAL array, dimension (LDC, N)

On entry, C contains the right-hand-side of the first matrix

equation in (1) or (3).

On exit, if IJOB = 0, 1 or 2, C has been overwritten by

the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,

the solution achieved during the computation of the

Dif-estimate.

*LDC*

LDC is INTEGER

The leading dimension of the array C. LDC >= max(1, M).

*D*

D is REAL array, dimension (LDD, M)

The upper triangular matrix D.

*LDD*

LDD is INTEGER

The leading dimension of the array D. LDD >= max(1, M).

*E*

E is REAL array, dimension (LDE, N)

The upper triangular matrix E.

*LDE*

LDE is INTEGER

The leading dimension of the array E. LDE >= max(1, N).

*F*

F is REAL array, dimension (LDF, N)

On entry, F contains the right-hand-side of the second matrix

equation in (1) or (3).

On exit, if IJOB = 0, 1 or 2, F has been overwritten by

the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,

the solution achieved during the computation of the

Dif-estimate.

*LDF*

LDF is INTEGER

The leading dimension of the array F. LDF >= max(1, M).

*DIF*

DIF is REAL

On exit DIF is the reciprocal of a lower bound of the

reciprocal of the Dif-function, i.e. DIF is an upper bound of

Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).

IF IJOB = 0 or TRANS = 'T', DIF is not touched.

*SCALE*

SCALE is REAL

On exit SCALE is the scaling factor in (1) or (3).

If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,

to a slightly perturbed system but the input matrices A, B, D

and E have not been changed. If SCALE = 0, C and F hold the

solutions R and L, respectively, to the homogeneous system

with C = F = 0. Normally, SCALE = 1.

*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.

If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).

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.

*IWORK*

IWORK is INTEGER array, dimension (M+N+6)

*INFO*

INFO is INTEGER

=0: successful exit

<0: If INFO = -i, the i-th argument had an illegal value.

>0: (A, D) and (B, E) have common or close eigenvalues.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**References:**

[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software

for Solving the Generalized Sylvester Equation and Estimating the

Separation between Regular Matrix Pairs, Report UMINF - 93.23,

Department of Computing Science, Umea University, S-901 87 Umea,

Sweden, December 1993, Revised April 1994, Also as LAPACK Working

Note 75. To appear in ACM Trans. on Math. Software, Vol 22,

No 1, 1996.

[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester

Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.

Appl., 15(4):1045-1060, 1994

[3] B. Kagstrom and L. Westin, Generalized Schur Methods with

Condition Estimators for Solving the Generalized Sylvester

Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,

July 1989, pp 745-751.

## subroutine strsyl (character TRANA, character TRANB, integer ISGN, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldc, * ) C, integer LDC, real SCALE, integer INFO)¶

**STRSYL**

**Purpose:**

STRSYL solves the real Sylvester matrix equation:

op(A)*X + X*op(B) = scale*C or

op(A)*X - X*op(B) = scale*C,

where op(A) = A or A**T, and A and B are both upper quasi-

triangular. A is M-by-M and B is N-by-N; the right hand side C and

the solution X are M-by-N; and scale is an output scale factor, set

<= 1 to avoid overflow in X.

A and B must be in Schur canonical form (as returned by SHSEQR), that

is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;

each 2-by-2 diagonal block has its diagonal elements equal and its

off-diagonal elements of opposite sign.

**Parameters**

*TRANA*

TRANA is CHARACTER*1

Specifies the option op(A):

= 'N': op(A) = A (No transpose)

= 'T': op(A) = A**T (Transpose)

= 'C': op(A) = A**H (Conjugate transpose = Transpose)

*TRANB*

TRANB is CHARACTER*1

Specifies the option op(B):

= 'N': op(B) = B (No transpose)

= 'T': op(B) = B**T (Transpose)

= 'C': op(B) = B**H (Conjugate transpose = Transpose)

*ISGN*

ISGN is INTEGER

Specifies the sign in the equation:

= +1: solve op(A)*X + X*op(B) = scale*C

= -1: solve op(A)*X - X*op(B) = scale*C

*M*

M is INTEGER

The order of the matrix A, and the number of rows in the

matrices X and C. M >= 0.

*N*

N is INTEGER

The order of the matrix B, and the number of columns in the

matrices X and C. N >= 0.

*A*

A is REAL array, dimension (LDA,M)

The upper quasi-triangular matrix A, in Schur canonical form.

*LDA*

LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,M).

*B*

B is REAL array, dimension (LDB,N)

The upper quasi-triangular matrix B, in Schur canonical form.

*LDB*

LDB is INTEGER

The leading dimension of the array B. LDB >= max(1,N).

*C*

C is REAL array, dimension (LDC,N)

On entry, the M-by-N right hand side matrix C.

On exit, C is overwritten by the solution matrix X.

*LDC*

LDC is INTEGER

The leading dimension of the array C. LDC >= max(1,M)

*SCALE*

SCALE is REAL

The scale factor, scale, set <= 1 to avoid overflow in X.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value

= 1: A and B have common or very close eigenvalues; perturbed

values were used to solve the equation (but the matrices

A and B are unchanged).

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

# Author¶

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