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converts sparse matrix into adjacency form

### Syntax

`[xadj,iadj,v]=sp2adj(A)`

### Arguments

A

m-by-n real or complex sparse matrix (with nz non-zero entries)

a (n+1)-by-1 matrix of floating point integers, pointers to the starting index in iadj and v for each column. For `j=1:n`, the floating point integer `xadj(j+1)-xadj(j)` is the number of non zero entries in column j.

a nz-by-1 matrix of floating point integers, the row indices for the nonzeros. For `j=1:n`, for `k = xadj(j):xadj(j+1)-1`, the floating point integer `i = iadj(k)` is the row index of the nonzero entry #k.

v

a nz-by-1 matrix of floating point integers, the non-zero entries of A. For `j=1:n`, for `k = xadj(j):xadj(j+1)-1`, the floating point integer `Aij = v(k)` is the value of the nonzero entry #k.

### Description

sp2adj converts a sparse matrix into its adjacency format. The values in the adjacency format are stored column-by-column. This is why this format is sometimes called "Compressed sparse column" or CSC.

### Examples

In the following example, we create a full matrix, which entries goes from 1 to 10. Then we convert it into a sparse matrix, which removes the zeros. Finally, we compute the adjacency representation of this matrix. The matrix v contains only the nonzero entries of A.

```A = [
0 0 4 0 9
0 0 5 0 0
1 3 0 7 0
0 0 6 0 10
2 0 0 8 0
];
B=sparse(A);
expected_xadj = [1 3 4 7 9 11]';
expected_adjncy = [3 5 3 1 2 4 3 5 1 4]';
expected_anz = [1 2 3 4 5 6 7 8 9 10]';
and(expected_anz == v) // Should be %t
// j is the column index
for j = 1 : size(xadj,"*")-1
mprintf("Column #%d:\n",j)
mprintf("  Rows  = %s:\n",sci2exp(irows))
mprintf("  Values= %s:\n",sci2exp(vcolj))
end```

The previous script produces the following output.

```Column #1:
Rows  = [3;5]:
Values= [1;2]:
Column #2:
Rows  = 3:
Values= 3:
Column #3:
Rows  = [1;2;4]:
Values= [4;5;6]:
Column #4:
Rows  = [3;5]:
Values= [7;8]:
Column #5:
Rows  = [1;4]:
Values= [9;10]:```

Let us consider the column #1. The equality xadj(2)-xadj(1)=2 indicates that there are two nonzeros in the column #1. The row indices are stored in iadj, which tells us that the nonzero entries in column #1 are at rows #3 and #5. The v matrix tells us the actual entries are equal to 1 and 2.

In the following example, we browse the nonzero entries of a sparse matrix by looping on the adjacency structure.

```A = [
0 0 0 0 0 6 0 0 0 0
3 0 5 0 0 0 0 5 0 0
0 0 0 3 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 7 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 3
0 0 0 0 0 0 0 0 2 0
];
B=sparse(A);
expected_xadj = [1 2 3 4 5 5 6 6 7 8 9]';
expected_adjncy = [2 5 2 3 1 2 7 6]';
expected_anz = [3 7 5 3 6 5 2 3]';
and(expected_anz == v) // Should be %t```

In the following example, we check that the sp2adj and adj2sp functions are inverse.

```// Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide
// Edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst
// "Sparse Matrix Storage Formats", J. Dongarra
// http://web.eecs.utk.edu/~dongarra/etemplates/book.html

A = [
10 0 0 0 -2 0
3 9 0 0 0 3
0 7 8 7 0 0
3 0 8 7 5 0
0 8 0 9 9 13
0 4 0 0 2 -1
];
A = sparse(A)

// To get the Compressed Sparse Column (CSC) :
// To convert back to sparse:
// Check the conversion
AAsp - A

// To get the Compressed Sparse Row (CSR) :
// To convert back to sparse:
// Check the conversion
AAsp - A```

• sparse — sparse matrix definition
• spcompack — converts a compressed adjacency representation
• spget — retrieves entries of sparse matrix

### References

"Implementation of Lipsol in Scilab", Hector E. Rubio Scola, INRIA, Decembre 1997, Rapport Technique No 0215

"Solving Large Linear Optimization Problems with Scilab : Application to Multicommodity Problems", Hector E. Rubio Scola, Janvier 1999, Rapport Technique No 0227

"Toolbox Scilab : Detection signal design for failure detection and isolation for linear dynamic systems User's Guide", Hector E. Rubio Scola, 2000, Rapport Technique No 0241

"Computer Solution of Large Sparse Positive Definite Systems", A. George, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1981.