## table of contents

doubleSYsolve(3) | LAPACK | doubleSYsolve(3) |

# NAME¶

doubleSYsolve

# SYNOPSIS¶

## Functions¶

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

** DSYSV computes the solution to system of linear equations A * X = B for SY
matrices** subroutine **dsysv_aa** (UPLO, N, NRHS, A, LDA, IPIV, B,
LDB, WORK, LWORK, INFO)

** DSYSV_AA computes the solution to system of linear equations A * X = B for
SY matrices** subroutine **dsysv_aa_2stage** (UPLO, N, NRHS, A, LDA,
TB, LTB, IPIV, IPIV2, B, LDB, WORK, LWORK, INFO)

** DSYSV_AA_2STAGE computes the solution to system of linear equations A * X
= B for SY matrices** subroutine **dsysv_rk** (UPLO, N, NRHS, A, LDA,
E, IPIV, B, LDB, WORK, LWORK, INFO)

** DSYSV_RK computes the solution to system of linear equations A * X = B for
SY matrices** subroutine **dsysv_rook** (UPLO, N, NRHS, A, LDA, IPIV,
B, LDB, WORK, LWORK, INFO)

** DSYSV_ROOK computes the solution to system of linear equations A * X = B
for SY matrices** subroutine **dsysvx** (FACT, UPLO, N, NRHS, A, LDA,
AF, LDAF, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, IWORK, INFO)

** DSYSVX computes the solution to system of linear equations A * X = B for
SY matrices** subroutine **dsysvxx** (FACT, UPLO, N, NRHS, A, LDA, AF,
LDAF, IPIV, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS,
ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)

**DSYSVXX**

# Detailed Description¶

This is the group of double solve driver functions for SY matrices

# Function Documentation¶

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

** DSYSV computes the solution to system of linear equations A *
X = B for SY matrices**

**Purpose:**

DSYSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric matrix and X and B are N-by-NRHS

matrices.

The diagonal pivoting method is used to factor A as

A = U * D * U**T, if UPLO = 'U', or

A = L * D * L**T, if UPLO = 'L',

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. The factored form of A is then

used to solve the system of equations A * X = B.

**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.

*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 DOUBLE PRECISION 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 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

DSYTRF.

*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 DSYTRF. 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.

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B.

On exit, if INFO = 0, the N-by-NRHS solution matrix X.

*LDB*

LDB is INTEGER

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

*WORK*

WORK is DOUBLE PRECISION 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, and for best performance

LWORK >= max(1,N*NB), where NB is the optimal blocksize for

DSYTRF.

for LWORK < N, TRS will be done with Level BLAS 2

for LWORK >= N, TRS will be done with Level BLAS 3

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, so the solution could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

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

** DSYSV_AA computes the solution to system of linear equations A
* X = B for SY matrices**

**Purpose:**

DSYSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric matrix and X and B are N-by-NRHS

matrices.

Aasen's algorithm is used to factor A as

A = U**T * T * U, if UPLO = 'U', or

A = L * T * L**T, if UPLO = 'L',

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

triangular matrices, and T is symmetric tridiagonal. The factored

form of A is then used to solve the system of equations A * X = B.

**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.

*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 DOUBLE PRECISION 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 tridiagonal matrix T and the

multipliers used to obtain the factor U or L from the

factorization A = U**T*T*U or A = L*T*L**T as computed by

DSYTRF.

*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).

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B.

On exit, if INFO = 0, the N-by-NRHS solution matrix X.

*LDB*

LDB is INTEGER

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

*WORK*

WORK is DOUBLE PRECISION 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,3*N-2), and for

the best performance, LWORK >= MAX(1,N*NB), where NB is

the optimal blocksize for DSYTRF_AA.

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, so the solution could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine dsysv_aa_2stage (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TB, integer LTB, integer, dimension( * ) IPIV, integer, dimension( * ) IPIV2, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

** DSYSV_AA_2STAGE computes the solution to system of linear
equations A * X = B for SY matrices**

**Purpose:**

DSYSV_AA_2STAGE computes the solution to a real system of

linear equations

A * X = B,

where A is an N-by-N symmetric matrix and X and B are N-by-NRHS

matrices.

Aasen's 2-stage algorithm is used to factor A as

A = U**T * T * U, if UPLO = 'U', or

A = L * T * L**T, if UPLO = 'L',

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

triangular matrices, and T is symmetric and band. The matrix T is

then LU-factored with partial pivoting. The factored form of A

is then used to solve the system of equations A * X = B.

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.

*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 DOUBLE PRECISION 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 DOUBLE PRECISION 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).

*B*

B is DOUBLE PRECISION 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 DOUBLE PRECISION 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.

**Date**

## subroutine dsysv_rk (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

** DSYSV_RK computes the solution to system of linear equations A
* X = B for SY matrices**

**Purpose:**

DSYSV_RK computes the solution to a real system of linear

equations A * X = B, where A is an N-by-N symmetric matrix

and X and B are N-by-NRHS matrices.

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used

to factor A as

A = P*U*D*(U**T)*(P**T), if UPLO = 'U', or

A = P*L*D*(L**T)*(P**T), if UPLO = 'L',

where U (or L) is unit upper (or lower) triangular matrix,

U**T (or L**T) is the transpose of U (or L), P is a permutation

matrix, P**T is the transpose of P, and D is symmetric and block

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

DSYTRF_RK is called to compute the factorization of a real

symmetric matrix. The factored form of A is then used to solve

the system of equations A * X = B by calling BLAS3 routine DSYTRS_3.

**Parameters**

*UPLO*

UPLO is CHARACTER*1

Specifies whether the upper or lower triangular part of the

symmetric matrix A is stored:

= '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. NRHS >= 0.

*A*

A is DOUBLE PRECISION 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, diagonal of the block diagonal

matrix D and factors U or L as computed by DSYTRF_RK:

a) ONLY diagonal elements of the symmetric block diagonal

matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);

(superdiagonal (or subdiagonal) elements of D

are stored on exit in array E), and

b) If UPLO = 'U': factor U in the superdiagonal part of A.

If UPLO = 'L': factor L in the subdiagonal part of A.

For more info see the description of DSYTRF_RK routine.

*LDA*

LDA is INTEGER

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

*E*

E is DOUBLE PRECISION array, dimension (N)

On exit, contains the output computed by the factorization

routine DSYTRF_RK, i.e. the superdiagonal (or subdiagonal)

elements of the symmetric block diagonal matrix D

with 1-by-1 or 2-by-2 diagonal blocks, where

If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;

If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.

NOTE: For 1-by-1 diagonal block D(k), where

1 <= k <= N, the element E(k) is set to 0 in both

UPLO = 'U' or UPLO = 'L' cases.

For more info see the description of DSYTRF_RK routine.

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D,

as determined by DSYTRF_RK.

For more info see the description of DSYTRF_RK routine.

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B.

On exit, if INFO = 0, the N-by-NRHS solution matrix X.

*LDB*

LDB is INTEGER

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

*WORK*

WORK is DOUBLE PRECISION array, dimension ( MAX(1,LWORK) ).

Work array used in the factorization stage.

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*

LWORK is INTEGER

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

of factorization stage LWORK >= max(1,N*NB), where NB is

the optimal blocksize for DSYTRF_RK.

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

the routine only calculates the optimal size of the WORK

array for factorization stage, 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 = -k, the k-th argument had an illegal value

> 0: If INFO = k, the matrix A is singular, because:

If UPLO = 'U': column k in the upper

triangular part of A contains all zeros.

If UPLO = 'L': column k in the lower

triangular part of A contains all zeros.

Therefore D(k,k) is exactly zero, and superdiagonal

elements of column k of U (or subdiagonal elements of

column k of L ) are all zeros. 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.

NOTE: INFO only stores the first occurrence of

a singularity, any subsequent occurrence of singularity

is not stored in INFO even though the factorization

always completes.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 dsysv_rook (character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) WORK, integer LWORK, integer INFO)¶

** DSYSV_ROOK computes the solution to system of linear equations
A * X = B for SY matrices**

**Purpose:**

DSYSV_ROOK computes the solution to a real system of linear

equations

A * X = B,

where A is an N-by-N symmetric matrix and X and B are N-by-NRHS

matrices.

The diagonal pivoting method is used to factor A as

A = U * D * U**T, if UPLO = 'U', or

A = L * D * L**T, if UPLO = 'L',

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.

DSYTRF_ROOK is called to compute the factorization of a real

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

pivoting method.

The factored form of A is then used to solve the system

of equations A * X = B by calling DSYTRS_ROOK.

**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.

*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 DOUBLE PRECISION 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 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

DSYTRF_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 DSYTRF_ROOK.

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.

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B.

On exit, if INFO = 0, the N-by-NRHS solution matrix X.

*LDB*

LDB is INTEGER

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

*WORK*

WORK is DOUBLE PRECISION 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, and for best performance

LWORK >= max(1,N*NB), where NB is the optimal blocksize for

DSYTRF_ROOK.

TRS will be done with Level 2 BLAS

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, so the solution could not be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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 dsysvx (character FACT, character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

** DSYSVX computes the solution to system of linear equations A *
X = B for SY matrices**

**Purpose:**

DSYSVX uses the diagonal pivoting factorization to compute the

solution to a real system of linear equations A * X = B,

where A is an N-by-N symmetric matrix and X and B are N-by-NRHS

matrices.

Error bounds on the solution and a condition estimate are also

provided.

**Description:**

The following steps are performed:

1. If FACT = 'N', the diagonal pivoting method is used to factor A.

The form of the factorization is

A = U * D * U**T, if UPLO = 'U', or

A = L * D * L**T, if UPLO = 'L',

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.

2. If some D(i,i)=0, so that D is exactly singular, then the routine

returns with INFO = i. Otherwise, the factored form of A is used

to estimate the condition number of the matrix A. If the

reciprocal of the condition number is less than machine precision,

INFO = N+1 is returned as a warning, but the routine still goes on

to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form

of A.

4. Iterative refinement is applied to improve the computed solution

matrix and calculate error bounds and backward error estimates

for it.

**Parameters**

*FACT*

FACT is CHARACTER*1

Specifies whether or not the factored form of A has been

supplied on entry.

= 'F': On entry, AF and IPIV contain the factored form of

A. AF and IPIV will not be modified.

= 'N': The matrix A will be copied to AF and factored.

*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 matrices B and X. NRHS >= 0.

*A*

A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDAF,N)

If FACT = 'F', then AF is an input argument and on entry

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 DSYTRF.

If FACT = 'N', then AF is an output argument and on exit

returns 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.

*LDAF*

LDAF is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

If FACT = 'F', then IPIV is an input argument and on entry

contains details of the interchanges and the block structure

of D, as determined by DSYTRF.

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.

If FACT = 'N', then IPIV is an output argument and on exit

contains details of the interchanges and the block structure

of D, as determined by DSYTRF.

*B*

B is DOUBLE PRECISION array, dimension (LDB,NRHS)

The N-by-NRHS right hand side matrix B.

*LDB*

LDB is INTEGER

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

*X*

X is DOUBLE PRECISION array, dimension (LDX,NRHS)

If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

*LDX*

LDX is INTEGER

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

*RCOND*

RCOND is DOUBLE PRECISION

The estimate of the reciprocal condition number of the matrix

A. If RCOND is less than the machine precision (in

particular, if RCOND = 0), the matrix is singular to working

precision. This condition is indicated by a return code of

INFO > 0.

*FERR*

FERR is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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(1,3*N), and for best

performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where

NB is the optimal blocksize for DSYTRF.

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 (N)

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, and i is

<= N: D(i,i) is exactly zero. The factorization

has been completed but the factor D is exactly

singular, so the solution and error bounds could

not be computed. RCOND = 0 is returned.

= N+1: D is nonsingular, but RCOND is less than machine

precision, meaning that the matrix is singular

to working precision. Nevertheless, the

solution and error bounds are computed because

there are a number of situations where the

computed solution can be more accurate than the

value of RCOND would suggest.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

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

**DSYSVXX**

**Purpose:**

DSYSVXX uses the diagonal pivoting factorization to compute the

solution to a double precision system of linear equations A * X = B, where A

is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

If requested, both normwise and maximum componentwise error bounds

are returned. DSYSVXX will return a solution with a tiny

guaranteed error (O(eps) where eps is the working machine

precision) unless the matrix is very ill-conditioned, in which

case a warning is returned. Relevant condition numbers also are

calculated and returned.

DSYSVXX accepts user-provided factorizations and equilibration

factors; see the definitions of the FACT and EQUED options.

Solving with refinement and using a factorization from a previous

DSYSVXX call will also produce a solution with either O(eps)

errors or warnings, but we cannot make that claim for general

user-provided factorizations and equilibration factors if they

differ from what DSYSVXX would itself produce.

**Description:**

The following steps are performed:

1. If FACT = 'E', double precision scaling factors are computed to equilibrate

the system:

diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B

Whether or not the system will be equilibrated depends on the

scaling of the matrix A, but if equilibration is used, A is

overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the LU decomposition is used to factor

the matrix A (after equilibration if FACT = 'E') as

A = U * D * U**T, if UPLO = 'U', or

A = L * D * L**T, if UPLO = 'L',

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.

3. If some D(i,i)=0, so that D is exactly singular, then the

routine returns with INFO = i. Otherwise, the factored form of A

is used to estimate the condition number of the matrix A (see

argument RCOND). If the reciprocal of the condition number is

less than machine precision, the routine still goes on to solve

for X and compute error bounds as described below.

4. The system of equations is solved for X using the factored form

of A.

5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),

the routine will use iterative refinement to try to get a small

error and error bounds. Refinement calculates the residual to at

least twice the working precision.

6. If equilibration was used, the matrix X is premultiplied by

diag(R) so that it solves the original system before

equilibration.

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**

*FACT*

FACT is CHARACTER*1

Specifies whether or not the factored form of the matrix A is

supplied on entry, and if not, whether the matrix A should be

equilibrated before it is factored.

= 'F': On entry, AF and IPIV contain the factored form of A.

If EQUED is not 'N', the matrix A has been

equilibrated with scaling factors given by S.

A, AF, and IPIV are not modified.

= 'N': The matrix A will be copied to AF and factored.

= 'E': The matrix A will be equilibrated if necessary, then

copied to AF and factored.

*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 matrices B and X. NRHS >= 0.

*A*

A is DOUBLE PRECISION 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.

On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by

diag(S)*A*diag(S).

*LDA*

LDA is INTEGER

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

*AF*

AF is DOUBLE PRECISION array, dimension (LDAF,N)

If FACT = 'F', then AF is an input argument and on entry

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 DSYTRF.

If FACT = 'N', then AF is an output argument and on exit

returns 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.

*LDAF*

LDAF is INTEGER

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

*IPIV*

IPIV is INTEGER array, dimension (N)

If FACT = 'F', then IPIV is an input argument and on entry

contains details of the interchanges and the block

structure of D, as determined by DSYTRF. 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.

If FACT = 'N', then IPIV is an output argument and on exit

contains details of the interchanges and the block

structure of D, as determined by DSYTRF.

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

= 'N': No equilibration (always true if FACT = 'N').

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

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

EQUED is an input argument if FACT = 'F'; otherwise, it is an

output argument.

*S*

S is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B.

On exit,

if EQUED = 'N', B is not modified;

if EQUED = 'Y', B is overwritten by diag(S)*B;

*LDB*

LDB is INTEGER

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

*X*

X is DOUBLE PRECISION array, dimension (LDX,NRHS)

If INFO = 0, the N-by-NRHS solution matrix X to the original

system of equations. Note that A and B are modified on exit if

EQUED .ne. 'N', and the solution to the equilibrated system is

inv(diag(S))*X.

*LDX*

LDX is INTEGER

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

*RCOND*

RCOND is DOUBLE PRECISION

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.

*RPVGRW*

RPVGRW is DOUBLE PRECISION

Reciprocal pivot growth. On exit, this contains the reciprocal

pivot growth factor norm(A)/norm(U). The "max absolute element"

norm is used. If this is much less than 1, then 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. If factorization fails with

0<INFO<=N, then this contains the reciprocal pivot growth factor

for the leading INFO columns of A.

*BERR*

BERR is DOUBLE PRECISION 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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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: 1.0D+0

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

computed.

= 1.0: Use the extra-precise refinement algorithm.

(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 DOUBLE PRECISION 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.

**Date**

# Author¶

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