Work Text:
Strictly Diagonally Dominant (SDD) is a class of matrices that is frequently encountered in some applications. A matrix is SDD when the absolute value of each of its diagonal entries is bigger than the sum of all the absolute values of the other entries in that diagonal entry’s row. For example, a matrix consisting of
7 2 2
2 5 1
1 2 6
is SDD. This type of matrix has a few convenient properties: if a matrix is SDD, then
(1) performing Gaussian Elimination on the matrix preserves its SDD property.
(2) it is invertible and can be factored as A = LU (see Factorization).
(3) Gaussian Elimination with scaled partial pivoting will not require row interchanges.
Additionally, if a matrix is SDD and symmetric, then Gaussian Elimination with partial pivoting will not require row interchanges.
***
Sherlock and John are arranged as such:
John’s bed forms a frame around them. Them, both of them, because they are both in the bed, because two hours ago Sherlock brought home Chinese takeaway.
John and Sherlock are naked. Completely naked, clothes strewn over the floor and twisted in with the disorderly bedsheets like mangled intestines, because two hours ago Sherlock brought home Chinese takeaway, because Sherlock was an impossible git.
Sherlock’s toes dangle off the corner of the bed. Sherlock, being Sherlock, must lie diagonally across the bed, with the way his arms are stretched out past his head, if he wants to fit on, and he does want to fit on, because two hours ago Sherlock brought home Chinese takeaway, because Sherlock was an impossible git and felt the need to apologize for once.
John’s legs lie parallel on the outsides of Sherlock’s, or almost parallel, anyway, splayed slightly outward to accommodate the bend of his knees. John fits on the bed perfectly fine, thank you, but he lines up diagonally with Sherlock because he must, or because he really, really needs to right now, because two hours ago Sherlock brought home Chinese takeaway, because Sherlock was an impossible git and felt the need to apologize for once for the stupid mistake he made today at Tesco.
John kisses the back of Sherlock’s neck, which is directly beneath him. Christ is all he can think as he does so, that and god and fuck and yes and Sherlock, Sherlock, Sherlock. His nose tickles as it digs into stray curls and he inhales deeply, for Sherlock still smells strongly of sweat, because two hours ago Sherlock brought home Chinese takeaway, because Sherlock was an impossible git and felt the need to apologize for once for the stupid mistake he made today at Tesco just as John was coming home from his locum work.
“John,” Sherlock groans into the pillow, curling his toes and raising his arse, which is also directly beneath John, which is also—Christ, god, fuck, yes, John, John, John—around John, with their bodies diagonal on the bed, with John kissing his neck and devouring the skin above segments of his spinal column. Sherlock bites his lip and his chest is still shaking and heaving, mostly thanks to Christ, god, fuck, yes, John, partly because two hours ago Sherlock brought home Chinese takeaway, because Sherlock was an impossible git and felt the need to apologize for once for the stupid mistake he made today at Tesco just as John was coming home from his locum work—and thank god for the sheer dumb luck of John’s timing.
As Sherlock raises his arse to take in more of John, John stutter-stop-gasps and Christ, god, fuck, yes digs his fingers into the intestinal twist of sheets and clothes, employs every remaining thread of his mental fortitude to keep from bucking wildly to finish himself off, because he can’t just do that, because this is new, very very new, and it’s just as new for Sherlock. They really, really hadn’t meant to go this far, not for a very long while, anyway, still stuck up until today on prolonged stares and John licking his lips and Sherlock swaying closer and both of them, surely, being sure that Sherlock was only moments away from leaning in and kissing John and John seconds from ripping the buttons from Sherlock’s shirt. They hadn’t meant to go this far but it all just burst because two hours ago Sherlock brought home Chinese takeaway, because Sherlock was an impossible git and felt the need to apologize for once for the stupid mistake he made today at Tesco just as John was coming home from his locum work—and thank god for the sheer dumb luck of John’s timing, and the fact that he happened to have stopped by Tesco on the way home and heard the commotion.
Sherlock huffs into the pillow and reaches down for his cock. “Please, please, please, John,” he says, grasping himself, which seems to startle John into action, and John braces himself on one elbow to thrust faster, his forearm running parallel to Sherlock’s side. With his other arm, John reaches forward to twist his fingers into Sherlock’s hair, to yank his head back slightly, to nip, to kiss, to make love to the back of his neck. Sherlock’s exposed throat bobs in pushing back against moans that escape him like whimpers, and for all the time he spent Married to his Work, and for all the time John spent being Not Gay, they are taking one another apart as if they have been from the start—they have been, haven’t they?—decomposing into their most basic components, only skin and sweat and primal sounds. To think, to think that this is only happening by chance, only happening because two hours ago Sherlock brought home Chinese takeaway, because Sherlock was an impossible git and felt the need to apologize for once for the stupid mistake he made today at Tesco just as John was coming home from his locum work—and thank god for the sheer dumb luck of John’s timing, and the fact that he happened to have stopped by Tesco on the way home and heard the commotion, saw Sherlock’s scarf tied like a repel line to the jutting metal of a sign on the side of the building and climbed, and found Sherlock, helpless, stuck, alone, stupid, without John because he apparently couldn’t wait a bloody half an hour for John to get home—Sherlock, staring down a burly man who’d drawn a gun and pointed it at him.
John lets the blinding heat of quickly pumping blood take his brains through the ceiling, fill his entire body with helium before his hips jerk, jerk, jerk into Sherlock and he crashes down with stars and sparks and barking and gasping and finally, whining, and lying along Sherlock’s back as from Sherlock’s lungs seeps a deep groan. Before the clothes and the bed and Sherlock’s apology and the Chinese takeaway, on the roof of their usual Tesco, John, of course, had his own gun on him—habit—and shot the man, the man who was about to shoot Sherlock, the man who’d shouted and threatened and had his gun loaded and ready and pointed and his finger on the trigger—John shot him in the knee.
Of course, the man had also fallen off the building as a result.
And they had called the police and John swept Sherlock away for Baker Street and shouted at Sherlock until his throat scratched because Sherlock had almost died and Sherlock was an impossible git and Sherlock, Sherlock, Sherlock, thank god, was alive, and John’s chest burst when he said it, when he said, “You could have died.” With John’s bursting chest came blurred eyes, came Sherlock’s long, gentle fingers to John’s hands, curled subtly around his wrists, came Sherlock’s lips pressed against John’s for just a moment. Sherlock told John he needed to dash back out to make sure Lestrade got some of the finer points, and left John alone to puzzle over his lips.
Sherlock came back half an hour later with Chinese takeaway. He came back and they ate and said nothing and then Sherlock said, “You’re right,” and then, “I’m sorry.”
John said, “Don’t do it again, or…” and his throat caught, and hitched, and, remembering his breathlessness and blurred eyes from before, had to fight them back again. He locked eyes with Sherlock and Sherlock stepped closer and John grabbed him—grabbed him and held him fast, pinned with strong arms against his body, and Sherlock leaned down and pressed his lips to John’s again. They had tumbled up the stairs, all hands and mouths, to John’s room; they had tumbled into the bed. They had stripped off clothes and wrestled in sheets and twisted them like intestines and grabbed and held and groaned, and Sherlock had said please, John and John obliged, and thought perhaps he could make Sherlock his enough that Sherlock would never forget that he needs John every bit as much as John needs Sherlock. John had buried his cock in Sherlock’s arse as they laid diagonally across the bed, Christ god fuck yes, and they sweated out every urgent message that needed to be said and then, afterward, bathed in the humidity, in silence.
“John,” Sherlock finally breathes, and breathes, as if that is all he can manage.
John kisses Sherlock’s neck. “I know.”