## table of contents

realOTHERsolve(3) | LAPACK | realOTHERsolve(3) |

# NAME¶

realOTHERsolve

# SYNOPSIS¶

## Functions¶

subroutine **sgglse** (M, N, P, A, LDA, B, LDB, C, D, X, WORK,
LWORK, INFO)

** SGGLSE solves overdetermined or underdetermined systems for OTHER
matrices** subroutine **spbsv** (UPLO, N, KD, NRHS, AB, LDAB, B, LDB,
INFO)

** SPBSV computes the solution to system of linear equations A * X = B for
OTHER matrices** subroutine **spbsvx** (FACT, UPLO, N, KD, NRHS, AB,
LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK,
INFO)

** SPBSVX computes the solution to system of linear equations A * X = B for
OTHER matrices** subroutine **sppsv** (UPLO, N, NRHS, AP, B, LDB, INFO)

** SPPSV computes the solution to system of linear equations A * X = B for
OTHER matrices** subroutine **sppsvx** (FACT, UPLO, N, NRHS, AP, AFP,
EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)

** SPPSVX computes the solution to system of linear equations A * X = B for
OTHER matrices** subroutine **sspsv** (UPLO, N, NRHS, AP, IPIV, B, LDB,
INFO)

** SSPSV computes the solution to system of linear equations A * X = B for
OTHER matrices** subroutine **sspsvx** (FACT, UPLO, N, NRHS, AP, AFP,
IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)

** SSPSVX computes the solution to system of linear equations A * X = B for
OTHER matrices**

# Detailed Description¶

This is the group of real Other Solve routines

# Function Documentation¶

## subroutine sgglse (integer M, integer N, integer P, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) C, real, dimension( * ) D, real, dimension( * ) X, real, dimension( * ) WORK, integer LWORK, integer INFO)¶

** SGGLSE solves overdetermined or underdetermined systems for
OTHER matrices**

**Purpose:**

SGGLSE solves the linear equality-constrained least squares (LSE)

problem:

minimize || c - A*x ||_2 subject to B*x = d

where A is an M-by-N matrix, B is a P-by-N matrix, c is a given

M-vector, and d is a given P-vector. It is assumed that

P <= N <= M+P, and

rank(B) = P and rank( (A) ) = N.

( (B) )

These conditions ensure that the LSE problem has a unique solution,

which is obtained using a generalized RQ factorization of the

matrices (B, A) given by

B = (0 R)*Q, A = Z*T*Q.

**Parameters**

*M*

M is INTEGER

The number of rows of the matrix A. M >= 0.

*N*

N is INTEGER

The number of columns of the matrices A and B. N >= 0.

*P*

P is INTEGER

The number of rows of the matrix B. 0 <= P <= N <= M+P.

*A*

A is REAL array, dimension (LDA,N)

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

On exit, the elements on and above the diagonal of the array

contain the min(M,N)-by-N upper trapezoidal matrix T.

*LDA*

LDA is INTEGER

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

*B*

B is REAL array, dimension (LDB,N)

On entry, the P-by-N matrix B.

On exit, the upper triangle of the subarray B(1:P,N-P+1:N)

contains the P-by-P upper triangular matrix R.

*LDB*

LDB is INTEGER

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

*C*

C is REAL array, dimension (M)

On entry, C contains the right hand side vector for the

least squares part of the LSE problem.

On exit, the residual sum of squares for the solution

is given by the sum of squares of elements N-P+1 to M of

vector C.

*D*

D is REAL array, dimension (P)

On entry, D contains the right hand side vector for the

constrained equation.

On exit, D is destroyed.

*X*

X is REAL array, dimension (N)

On exit, X is the solution of the LSE problem.

*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 >= max(1,M+N+P).

For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,

where NB is an upper bound for the optimal blocksizes for

SGEQRF, SGERQF, SORMQR and SORMRQ.

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.

= 1: the upper triangular factor R associated with B in the

generalized RQ factorization of the pair (B, A) is

singular, so that rank(B) < P; the least squares

solution could not be computed.

= 2: the (N-P) by (N-P) part of the upper trapezoidal factor

T associated with A in the generalized RQ factorization

of the pair (B, A) is singular, so that

rank( (A) ) < N; the least squares solution could not

( (B) )

be computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine spbsv (character UPLO, integer N, integer KD, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

** SPBSV computes the solution to system of linear equations A *
X = B for OTHER matrices**

**Purpose:**

SPBSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric positive definite band matrix and X

and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as

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

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

where U is an upper triangular band matrix, and L is a lower

triangular band matrix, with the same number of superdiagonals or

subdiagonals as A. 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.

*KD*

KD is INTEGER

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

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

*NRHS*

NRHS is INTEGER

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

of the matrix B. NRHS >= 0.

*AB*

AB is REAL array, dimension (LDAB,N)

On entry, 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).

See below for further details.

On exit, if INFO = 0, the triangular factor U or L from the

Cholesky factorization A = U**T*U or A = L*L**T of the band

matrix A, in the same storage format as A.

*LDAB*

LDAB is INTEGER

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

*B*

B is REAL 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).

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, the leading minor of order i of A is not

positive definite, so the factorization could not be

completed, and the solution has not been computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

The band storage scheme is illustrated by the following example, when

N = 6, KD = 2, and UPLO = 'U':

On entry: On exit:

* * a13 a24 a35 a46 * * u13 u24 u35 u46

* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56

a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry: On exit:

a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66

a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *

a31 a42 a53 a64 * * l31 l42 l53 l64 * *

Array elements marked * are not used by the routine.

## subroutine spbsvx (character FACT, character UPLO, integer N, integer KD, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, character EQUED, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** SPBSVX computes the solution to system of linear equations A *
X = B for OTHER matrices**

**Purpose:**

SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to

compute the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric positive definite band 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 = 'E', real 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 Cholesky decomposition is used to

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

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

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

where U is an upper triangular band matrix, and L is a lower

triangular band matrix.

3. If the leading i-by-i principal minor is not positive definite,

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.

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

of A.

5. Iterative refinement is applied to improve the computed solution

matrix and calculate error bounds and backward error estimates

for it.

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

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

equilibration.

**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, AFB contains the factored form of A.

If EQUED = 'Y', the matrix A has been equilibrated

with scaling factors given by S. AB and AFB will not

be modified.

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

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

copied to AFB 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.

*KD*

KD is INTEGER

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

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

*AB*

AB is REAL array, dimension (LDAB,N)

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

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

if FACT = 'F' and EQUED = 'Y', then A must contain the

equilibrated matrix diag(S)*A*diag(S). 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).

See below for further details.

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

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

*LDAB*

LDAB is INTEGER

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

*AFB*

AFB is REAL array, dimension (LDAFB,N)

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

contains the triangular factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T of the band matrix

A, in the same storage format as A (see AB). If EQUED = 'Y',

then AFB is the factored form of the equilibrated matrix A.

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

returns the triangular factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T.

If FACT = 'E', then AFB is an output argument and on exit

returns the triangular factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T of the equilibrated

matrix A (see the description of A for the form of the

equilibrated matrix).

*LDAFB*

LDAFB is INTEGER

The leading dimension of the array AFB. LDAFB >= KD+1.

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

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

= 'Y': Equilibration was done, 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 REAL array, dimension (N)

The scale factors for A; not accessed if EQUED = 'N'. 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.

*B*

B is REAL 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 REAL array, dimension (LDX,NRHS)

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

the original system of equations. Note that if EQUED = 'Y',

A and B are modified on exit, 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 REAL

The estimate of the reciprocal condition number of the matrix

A after equilibration (if done). 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 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

> 0: if INFO = i, and i is

<= N: the leading minor of order i of A is

not positive definite, so the factorization

could not be completed, and the solution has not

been computed. RCOND = 0 is returned.

= N+1: U 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**

**Further Details:**

The band storage scheme is illustrated by the following example, when

N = 6, KD = 2, and UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13

a22 a23 a24

a33 a34 a35

a44 a45 a46

a55 a56

(aij=conjg(aji)) a66

Band storage of the upper triangle of A:

* * a13 a24 a35 a46

* a12 a23 a34 a45 a56

a11 a22 a33 a44 a55 a66

Similarly, if UPLO = 'L' the format of A is as follows:

a11 a22 a33 a44 a55 a66

a21 a32 a43 a54 a65 *

a31 a42 a53 a64 * *

Array elements marked * are not used by the routine.

## subroutine sppsv (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

** SPPSV computes the solution to system of linear equations A *
X = B for OTHER matrices**

**Purpose:**

SPPSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric positive definite matrix stored in

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

The Cholesky decomposition is used to factor A as

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

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

where U is an upper triangular matrix and L is a lower triangular

matrix. 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.

*AP*

AP is REAL array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix

A, packed columnwise in a linear array. The j-th column of A

is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

See below for further details.

On exit, if INFO = 0, the factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T, in the same storage

format as A.

*B*

B is REAL 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).

*INFO*

INFO is INTEGER

= 0: successful exit

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

> 0: if INFO = i, the leading minor of order i of A is not

positive definite, so the factorization could not be

completed, and the solution has not been computed.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**Further Details:**

The packed storage scheme is illustrated by the following example

when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13 a14

a22 a23 a24

a33 a34 (aij = conjg(aji))

a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

## subroutine sppsvx (character FACT, character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( * ) AFP, character EQUED, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** SPPSVX computes the solution to system of linear equations A *
X = B for OTHER matrices**

**Purpose:**

SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to

compute the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric positive definite matrix stored in

packed format 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 = 'E', real 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 Cholesky decomposition is used to

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

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

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

where U is an upper triangular matrix and L is a lower triangular

matrix.

3. If the leading i-by-i principal minor is not positive definite,

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.

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

of A.

5. Iterative refinement is applied to improve the computed solution

matrix and calculate error bounds and backward error estimates

for it.

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

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

equilibration.

**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, AFP contains the factored form of A.

If EQUED = 'Y', the matrix A has been equilibrated

with scaling factors given by S. AP and AFP will not

be modified.

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

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

copied to AFP 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.

*AP*

AP is REAL array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix

A, packed columnwise in a linear array, except if FACT = 'F'

and EQUED = 'Y', then A must contain the equilibrated matrix

diag(S)*A*diag(S). The j-th column of A is stored in the

array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

See below for further details. A is not modified if

FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.

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

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

*AFP*

AFP is REAL array, dimension (N*(N+1)/2)

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

contains the triangular factor U or L from the Cholesky

factorization A = U**T*U or A = L*L**T, in the same storage

format as A. If EQUED .ne. 'N', then AFP is the factored

form of the equilibrated matrix A.

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

returns the triangular factor U or L from the Cholesky

factorization A = U**T * U or A = L * L**T of the original

matrix A.

If FACT = 'E', then AFP is an output argument and on exit

returns the triangular factor U or L from the Cholesky

factorization A = U**T * U or A = L * L**T of the equilibrated

matrix A (see the description of AP for the form of the

equilibrated matrix).

*EQUED*

EQUED is CHARACTER*1

Specifies the form of equilibration that was done.

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

= 'Y': Equilibration was done, 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 REAL array, dimension (N)

The scale factors for A; not accessed if EQUED = 'N'. 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.

*B*

B is REAL 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 REAL array, dimension (LDX,NRHS)

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

the original system of equations. Note that if EQUED = 'Y',

A and B are modified on exit, 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 REAL

The estimate of the reciprocal condition number of the matrix

A after equilibration (if done). 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 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

> 0: if INFO = i, and i is

<= N: the leading minor of order i of A is

not positive definite, so the factorization

could not be completed, and the solution has not

been computed. RCOND = 0 is returned.

= N+1: U 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**

**Further Details:**

The packed storage scheme is illustrated by the following example

when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13 a14

a22 a23 a24

a33 a34 (aij = conjg(aji))

a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

## subroutine sspsv (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

** SSPSV computes the solution to system of linear equations A *
X = B for OTHER matrices**

**Purpose:**

SSPSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric matrix stored in packed format 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, 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.

*AP*

AP is REAL array, dimension (N*(N+1)/2)

On entry, the upper or lower triangle of the symmetric matrix

A, packed columnwise in a linear array. The j-th column of A

is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

See below for further details.

On exit, 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 SSPTRF, stored as

a packed triangular matrix in the same storage format as A.

*IPIV*

IPIV is INTEGER array, dimension (N)

Details of the interchanges and the block structure of D, as

determined by SSPTRF. 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 REAL 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).

*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**

**Further Details:**

The packed storage scheme is illustrated by the following example

when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13 a14

a22 a23 a24

a33 a34 (aij = aji)

a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

## subroutine sspsvx (character FACT, character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( * ) AFP, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

** SSPSVX computes the solution to system of linear equations A *
X = B for OTHER matrices**

**Purpose:**

SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or

A = L*D*L**T to compute the solution to a real system of linear

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

in packed format 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 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.

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, AFP and IPIV contain the factored form of

A. AP, AFP and IPIV will not be modified.

= 'N': The matrix A will be copied to AFP 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.

*AP*

AP is REAL array, dimension (N*(N+1)/2)

The upper or lower triangle of the symmetric matrix A, packed

columnwise in a linear array. The j-th column of A is stored

in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;

if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

See below for further details.

*AFP*

AFP is REAL array, dimension (N*(N+1)/2)

If FACT = 'F', then AFP 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 SSPTRF, stored as

a packed triangular matrix in the same storage format as A.

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

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 SSPTRF, stored as

a packed triangular matrix in the same storage format as A.

*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 SSPTRF.

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 SSPTRF.

*B*

B is REAL 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 REAL 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 REAL

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 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

> 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**

**Further Details:**

The packed storage scheme is illustrated by the following example

when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13 a14

a22 a23 a24

a33 a34 (aij = aji)

a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

# Author¶

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